cengage / g tewani maths solutions chapter circles || coordinate geometry … · 2018. 12. 8. ·...
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CENGAGE/GTEWANIMATHSSOLUTIONS
CHAPTERCIRCLES||COORDINATEGEOMETRY
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1
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Find the equation of the circle with radius 5 whose center lies on the x-axis andpassesthroughthepoint(2,3).
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2
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If the lines are diameters of the circle which passesthroughthepoint(2,6),thenfinditsequation.
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3
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Findtheequationofthecirclehavingcenterat(2,3)andwhichtouches
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4
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Find the equation of the circle having radius 5 and which touches lineatpoint(1,2).
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5
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If representstherealcirclewithnonzeroradius,thenfindthevaluesof
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x + y = 6andx + 2y = 4
x + y = 1
3x+ 4y − 11 = 0
x2 + y2 − 2x + 2ay + a + 3 = 0a.
6
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Find the image of the circle in the line
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7
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Theline istangenttothecircleatthepoint(2,5)andthecenterofthecirclelieson .Thenfindtheradiusofthecircle.
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8
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If the lines and are tangents toacircle, thenfindtheradiusofthecircle.
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9
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Iftheequation
representsacircle,thenfindthevaluesof
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10
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Apoint movesinsuchawaythattheratioofitsdistancefromtwocoplanarpointsisalwaysafixednumber .Then,identifythelocusofthepoint.
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x2 + y2 − 2x + 4y − 4 = 02x− 3y + 5 = 0
2x − y + 1 = 0x − 2y = 4
3x − 4y + 4 = 0 6x − 8y − 7 = 0
px2 + (2 − q)xy + 3y2 − 6qx+ 30y+ 6q = 0
pandq.
P( ≠ 1)
11
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Provethatthemaximumnumberofpointswithrationalcoordinatesonacirclewhosecenteris istwo.
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12
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Prove that for all values of , the locus of the point of intersection of the linesand isacircle.
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13
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Findthelengthofthechord alongtheline Alsofindtheanglethatthechordsubtendsatthecircumferenceofthelargersegment.
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14
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
LetA(-2,2)andB(2,-2)betwopointsABsubtendsanangleof atanypointsPintheplaneinsuchawaythatareaof is8squareunit,thennumberofpossibeposition(s)ofPis
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15
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Ifacirclepassesthroughthepoint ,thenfinditscenter.
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
(√3, 0)
θx cos θ+ y sin θ = a x sin θ − y cosθ = b
x2 + y2 − 4y = 0 x + y = 1.
45 ∘
ΔPAB
(0, 0), (a, 0)and(0, b)
(1, − 2), (4, − 3)
16 Find theequationof thecirclewhichpasses through thepointsandwhosecenterliesontheline
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17
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Show that a cyclic quadrilateral is formed by the lines and taken in order. Find the
equationofthecircumcircle.
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18
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Find theequationof the circle if the chordof the circle joining (1, 2) andsubtents atthecenterofthecircle.
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19
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Find the equation of the circle which passes through (1, 0) and (0, 1) and has itsradiusassmallaspossible.
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(1, − 2), (4, − 3)3x + 4y = 7.
5x+ 3y = 9, x = 3y, 2x = y x + 4y + 2 = 0
( − 3, 1)900
20
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If theabscissaandordinatesof twopoints are the roots of the equationsand , respectively, then find theequation
ofthecirclewith asdiameter.
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21
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Tangents are drawn to from the point Thenfindtheequationofthecircumcircleoftriangle
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22
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thepointonacirclenearesttothepoint isatadistanceof4unitsandthefarthestpointis(6,5).Thenfindtheequationofthecircle.
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23
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Findtheparametricformoftheequationofthecircle
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24
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Findthecenterofthecircle
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Findthelengthof intercept,thecircle makesonthe
PandQx2 + 2ax− b2 = 0 x2 + 2px − q2 = 0
PQ
PAandPB x2 + y2 = a2 P (x1, y1).PAB.
P(2, 1)
x2 + y2 + px + py = 0.
x = − 1 + 2 cosθ, y = 3 + 2sin θ
x2 + y2 + 10x − 6y + 9 = 0
25 x-axis.
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26
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If the intercepts of the variable circle on the x- and yl-axis are 2 units and 4 units,respectively,thenfindthelocusofthecenterofthevariablecircle.
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27
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Find the equation of the circle which touches both the axes and the straight lineinthefirstquadrantandliesbelowit.
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28
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Acircletouchesthey-axisatthepoint(0,4)andcutsthex-axisinachordoflength6units.Thenfindtheradiusofthecircle.
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29
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Find the equation of the circle which is touched by , has its center on thepositivedirectionof thex=axisandcutsoffachordof length2unitsalong the line
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of Circle
4x+ 3y = 6
y = x
√3y − x = 0
30
WithTypicalConditions
Findtheequationsofthecirclespassingthroughthepoint andtouchingthelines and
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31
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find the greatest distance of the point from the circle
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32
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find the points on the circle which are the farthestandnearesttothepoint
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33
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findthevaluesof forwhichthepoint liesinthelargersegmentofthe circle made by the chord whose equation is
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34
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find thenumberofpoint having integral coordinatessatisfying thecondition
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35
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Thecircle doesnottouchorintersectthecoordinateaxes,andthepoint(1,4)isinsidethecircle.Findtherangeofvalueof
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( − 4, 3)x + y = 2 x− y = 2
P(10, 7)x2 + y2 − 4x − 2y − 20 = 0
x2 + y2 − 2x + 4y − 20 = 0( − 5, 6).
α (α − 1, α + 1)x2 + y2 − x − y − 6 = 0
x + y − 2 = 0
(x, y)x2 + y2 < 25
x2 + y2 − 6x − 10y + k = 0k.
36
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find the area of the region in which the points satisfy the inequatiesand .
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37
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
If the line is a diameter of the circle ,thenfindthevalueof
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38
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Prove that the locusof thepoint thatmovessuch that thesumof thesquaresof itsdistancesfromthethreeverticesofatriangleisconstantisacircle.
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39
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find the number of integral values of for which is theequationof a circlewhose radiusdoes
notexceed5.
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
4 < x2 + y2 < 16 3x2 − y2 ≥ 0
x + 2by + 7 = 0 x2 + y2 − 6x+ 2y = 0b
λx2 + y2 + λx + (1 − λ)y + 5 = 0
40 Ifacirclewhosecenteris touchestheline ,thenfinditsradius.
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41
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Ifoneendof theadiameterof thecircle is (3,2),thenfindtheotherendofthediameter.
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42
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Prove that the locus of the centroid of the triangle whose vertices areand ,where isaparameter,iscircle.
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43
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find the equation of the circle which passes through the pointsandthecenterliesontheline
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44
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findtheradiusofthecircle
.
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(1, − 3) 3x − 4y − 5 = 0
2x2 + 2y2 − 4x − 8y + 2 = 0
(a cos t, a sin t), (b sin t, − b cos t), (1, 0) t
(3, − 2)and( − 2, 0) 2x − y = 3
(x − 5)(x − 1) + (y − 7)(y − 4)= 0
45
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findtheequationsof thecircleswhichpassthroughtheoriginandcutoffchordsoflength fromeachofthelines
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46
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findtheequationofthecirclewhichtouchesthex-axisandwhosecenteris(1,2).
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47
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findtheequationofthecirclewhichtouchesboththeaxesandtheline
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48
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findtheequationofthecirclewithcenterat andwhichcutsoffaninterceptoflength6fromtheline
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49
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findthelocusofthecenterofthecirclewhichcutsoffinterceptsoflengthsfromthex-andthey-axis,respectively.
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a y = xandy = − x
x = c
(3, − 1)2x − 5y + 18 = 0
2aand2b
50
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Circlesaredrawnthroughthepoint (2,0) tocut interceptof length5unitson thex-axis.Iftheircenterslieinthefirstquadrant,thenfindtheirequation.
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51
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find the equation of the circle passing through the origin and cutting intercepts oflengths3unitsand4unitssfromthepositiveexes.
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52
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Find thepointof intersectionof thecircle with thex-axis.
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53
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Findthevaluesof forwhichthepoints lieonacircle.
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54
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Ifoneendofthediameteris(1,1)andtheotherendliesontheline ,thenfindthelocusofthecenterofthecircle.
x2 + y2 − 3x− 4y + 2 = 0
k (2k, 3k), (1, 0), (0, 1), and(0, 0)
x + y = 3
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55
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Tangent drawn from the point to the circle touches it at thepoint in the first quadrant. Find the coordinates of another point on the circlesuchthat .
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56
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
If the joinof and makesonobtuseangle at thenprovethan
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57
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
Find the range of values of for which the line cuts the circleatdistinctorcoincidentpoints.
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58
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
Ifthelines and cutthecoordinaeaxesatconcyclicpoints,thenprovethat
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59
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
AlineisdrawnthroughafixpointP( )tocutthecircle atAandB.ThenPA.PBisequalto:
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P(4, 0) x2 + y2 = 8A BAB = 4
(x1, y1) (x2, y2) (x3, y3),
(x3 − x1)(x3 − x2) + (y3 − y1)(y3− y2) < 0
m y = mx + 2x2 + y2 = 1
a1x + b1y + c1 = 0 a2x+ b2y + c2 = 0|a1a2| = |b1b2|
α, β x2 + y2 = r2
60
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
Two circles intersect at two distinct points in a line passingthrough meetscircles at ,respectively.Let bethemidpointof
meetscircles at , respectively.Thenprovethatisthemidpointof
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61
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Find the angle between the two tangents from the origin to the circle
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62
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Findtheequationsofthetangentstothecircle whichareparalleltothestraightline
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63
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Find the equation of the tangent at the endpoints of the diameter of circlewhichisinclinedatanangle withthepositivex-axis.
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
C1andC2 PandQP C1andC2 AandB Y
AB, andQY C1andC2 XandZ YXZ .
(x − 7)2 + (y + 1)2 = 25
x2 + y2 − 6x + 4y = 124x + 3y + 5 = 0
(x − a)2 + (y − b)2 = r2 θ
2 2
64 The tangent to the circle at also touches the circle . Find the coordinats of the corresponding point of
contact.
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65
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Provethattheequationofanytangenttothecircle isoftheform
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66
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
If thenfindthepositivevalueof forwhich isacommontangentto and
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67
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Provethattheanglebetweenthetangentsfrom tothecircle is
,where
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68
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Thelengthsofthetangentsfrom and toacircleare and ,respectively. Then, find the length of the tangent from to the samecircle.
x2 + y2 = 5 (1, − 2)x2 + y2 − 8x + 6y + 20 = 0
x2 + y2 − 2x + 4y − 4 = 0y = m(x− 1) + 3√1 + m2 − 2.
a > 2b > 0, m y = mx − b√1 + m2
x2 + y2 = b2 (x − a)2 + y2 = b2.
(α, β) x2 + y2 = a2
2 tan−1( )a
√S1S1 = α2 + β2 − a2
P(1, − 1) Q(3, 3) √2 √6R( − 1, − 5)
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69
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
If from any point on the circle tangents aredrawntothecircle
,thenfindtheanglebetweenthetangents.
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70
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Ifthedistancesfromtheoriginofthecentersofthreecircles
areinGP,thenprovethatthelengthsofthetangentsdrawntothemfromanypointonthecircle areinGP.
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71
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Findtheequationof thenormalstothecircle at thepointwhoseordinateis
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72
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Find theequationof thenormal to thecircle parallel to the line
P x2 + y2 + 2gx+ 2fy + c = 0,
x2 + y2 + 2gx + 2fy + c sin2α
+ (g 2 + f2)cos2α = 0
x2 + y2 + 2λx − c2 = 0,(i = 1, 2, 3),
x2 + y2 = c2
x2 + y2 − 8x − 2y + 12 = 0−1
x2 + y2 − 2x = 0x + 2y = 3.
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73
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Find the equation of the tangent to the circle whichmakesequalinterceptsonthepositivecoordinatesaxes.
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74
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
If the length tangent drawn from the point (5, 3) to the circleis7,thenfindthevalueof
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75
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
A pair of tangents are drawn from the origin to the circle.Thenfinditsequations.
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76
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Findtheequationofthenormaltothecircle atthepoint
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77
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Find the equations of tangents to the circle whichareperpendiculartotheline
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x2 + y2 + 4x − 4y + 4 = 0
x2 + y2 + 2x + ky + 17 = 0 k.
x2 + y2 + 20(x + y) + 20 = 0
x2 + y2 = 9 ( , )1
√2
1
√2
x2 + y2 − 22x − 4y + 25 = 05x + 12y + 8 = 0
78
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
An infinite number of tangents can be drawn from to the circle.Thenfindthevalueof
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79
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
If thecircle intersects the line at twodistinctpoints,thenfindthevaluesof
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80
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
If a line passing through theorigin touches the circle ,thenfinditsslope.
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81
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Thetangentatanypoint onthecircle meetsthecoordinateaxesat.Thenfindthelocusofthemidpointof
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82
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Findthelocusofapointwhichmovessothattheratioofthelengthsofthetangentstothecircles and is
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(1, 2)x2 + y2 − 2x − 4y + λ = 0 λ
x2 + y2 − 4x − 8y − 5 = 0 3x − 4y = mm.
(x− 4)2 + (y + 5)2 = 25
P x2 + y2 = 4AandB AB.
x2 + y2 + 4x+ 3 = 0 x2 + y2 − 6x + 5 = 0 2 : 3.
83
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Find the length of the tangent drawn from any point on the circletothecircle
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84
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Find the ratio of the length of the tangents from any point on the circle to the two circles
and
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85
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
If the chord of contact of the tangents drawn from a point on the circle to the circle touches the circle ,
thenprovethat and areinGP.
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86
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
If the straight line intersects the circle at point,thenfindthecoordinatesofthepointofintersectionofthetangentsdrawn
at tothecircle
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
x2 + y2 + 2gx + 2fy + c1 = 0 x2 + y2 + 2gx + 2fy + c2 = 0
15x2 + 15y2 − 48x+ 64y = 05x2 + 5y2 − 24x+ 32y + 75 = 0 5x2 + 5y2 − 48x + 64y + 300 = 0
x2 + y2 + y2 = a2 x2 + y2 = b2 x2 + y2 = c2
a, b c
x = 2y + 1 = 0 x2 + y2 = 25PandQPandQ x2 + y2 = 25.
( , )
87If the chord of contact of the tangents drawn from the point to the circle
subtendsarightangleatthecenter,thenprovethat
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88
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Tangents are drawn to from any arbitrary point on the line .Thecorrespondingchordofcontactpassesthroughafixedpoint
whosecoordinatesare (b) (d)
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89
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Findtheareaofthetriangleformedbythetangentsfromthepoint(4,3)tothecircleandthelinejoiningtheirpointsofcontact.
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90
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Find the equation of the chord of the circle whose middle point is
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91
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Avariablechordisdrawnthroughtheorigintothecircle .Findthelocusofthecenterofthecircledrawnonthischordasdiameter.
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(h, k)x2 + y2 = a2 h2 + k2 = 2a2.
x2 + y2 = 1 P2x+ y − 4 = 0
( , )12
12
( , 1)12
( , )12
14
(1, )12
x2 + y2 = 9
x2 + y2 = 9(1, − 2)
x2 + y2 − 2ax = 0
92
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Findtheequationofthechordofthecircle passingthroughthepoint(2,3)farthestfromthecenter.
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93
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Findthemiddlepointofthechordofthecircle interceptedontheline
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94
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Theline isthechordofcontactofthepoint withrespect
tothecircle , for(a) (b) (c)
(d)
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95
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Tangentsaredrawn to the circle at the pointswhere it ismet by thecircle .Finthepointofintersectionofthesetangents.
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
x2 + y2 = a2
x2 + y2 = 25x − 2y = 2
9x + y − 18 = 0 P(h, k)
2x2 + 2y2 − 3x + 5y − 7 = 0 ( , − )245
45
P(3, 1)
P( − 3, 1) ( − , )25
125
x2 + y2 = 9x2 + y2 + 3x + 4y + 2 = 0
96 The distance between the chords of contact of tangents to the circlefromtheorigin&thepoint(g,f)is
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97
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Find the number of common tangent to the circlesand
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98
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Findtheequationofacirclewithcenter(4,3)touchingthecircle
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99
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Find the condition if the circle whose equations are andtouchoneanotherexternally.
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100
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Findtheequationofthesmallcirclethattouchesthecircle andpassesthroughthepoint
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101
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Show that the circles andtoucheachother.Findthecoordinatesofthepoint
ofcontactandtheequationofthecommontangentatthepointofcontact.
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x2 + y2 + 2gx + 2fy + c = 0
x2 + y2 + 2x + 8y − 23 = 0x2 + y2 − 4x − 10y + 9 = 0
x2 + y2 = 1
x2 + y2 + c2 = 2axx2 + y2 + c2 − 2by = 0
x2 + y2 = 1(4, 3).
x2 + y2 − 10x+ 4y − 20 = 0x2 + y2 + 14x− 6y + 22 = 0
102
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Find the equations to the common tangents of the circlesand
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103
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
If the circles and intersect othrogonally, then prove that +
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104
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
A circle passes through the origin and has its center on If it cutsorthogonally,thenfindtheequationofthecircle.
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105
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
If the radical axis of the circles andtouchesthecircle ,show
thateither or
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Theequationofthreecirclesaregiven
x2 + y2 − 2x − 6y + 9 = 0 x2 + y2 + 6x − 2y + 1 = 0
x2 + y2 + 2a ′x + 2b ′y + c ′ = 02x2 + 2y2 + 2ax + 2by + c = 0 aa ′ bb
' = c + .c ′
2
y = xx2 + y2 − 4x − 6y + 10 = −
x2 + y2 + 2gx + 2fy + c = 02x2 + 2y2 + 3x + 8y + 2c = 0 x2 + y2 + 2x + 1 = 0
g =34
f = 2
106
.Determinethecoordinatesofthepoint suchthatthetangentsdrawnfromittothecircleareequalinlength.
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107
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Findtheequationofthecirclewhichcutsthethreecircles
and orthogonally.
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108
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Ifthetangentsaredrawntothecircle atthepointwhereitmeetsthecircle then find the point of intersection of thesetangents.
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109
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Find theanglewhich thecommonchordof andsubtendsattheorigin.
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Find the length of the common chord of the circles and
x2 + y2 = 1, x2 + y2 − 8x + 15
= 0, x2 + y2 + 10y + 24 = 0P
x2 + y2 − 3x − 6y + 14 = 0, x2
+ y2 − x − 4y + 8 = 0,x2 + y2 + 2x − 6y + 9 = 0
x2 + y2 = 12x2 + y2 − 5x + 3y − 2 = 0,
x2 + y2 − 4x = 0 x2 + y2 = 16
x2 + y2 + 2x + 6y = 02 2
110
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111
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If thecircle bisects thecircumferenceof thecirclethenprovethat
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112
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Provethatthepairofstraightlinesjoiningtheorigintothepointsofintersectionofthecircles and is
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113
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If the circle is completely contained in the circle,thenfindthevaluesof
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If istheanglebetweenthetworadii(onetoeachcircle)drawnfromoneofthepoint
x2 + y2 − 4x − 2y − 6 = 0
x2 + y2 + 2gx + 2fy + c = 0x2 + y2 + 2g ′ x+ 2f ′ y + c ′ = 02g ′(g − g ′) + 2f ′ (f − f ′ ) = c
− c'
x2 + y2 = a x2 + y2 + 2(gx + fy) = 0a ′ (x2 + y2) − 4(gx+ fy)2 = 0
x2 + y2 = 1x2 + y2 + 4x + 3y + k = 0 k.
θ2 2 2 2 2 2
114 of intersection of two circles and then prove
thatthelengthofthecommonchordofthetwocirclesis
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115
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Thecircles and toucheachotherexternallytoucheachotherinternallyintersectattwopointsnoneofthese
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116
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If the circles andintersects at points P and Q , then find the values of for which the line
passesthrough
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117
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Which of the following is a point on the common chord of the circle and (b)
(d)
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118
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Considerthecircles
They are such that these circles touch each other one of these circles lies entirelyinsidetheothereachofthesecirclesliesoutsidetheothertheyintersectattwopoints.
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119
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If thecirclesofsameradius andcentersat (2,3)and5,6)cutorthogonally, thenfind
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x2 + y2 = a2 (x − c)2 + y2 = b2,2ab sin θ
√a2 + b2 − 2ab cosθ
x2 + y2 − 12x − 12y = 0 x2 + y2 + 6x + 6y = 0.
x2 + y2 + 2ax + cy + a = 0 x2 + y2 − 3ax + dy − 1 = 0a
5x+ by − a = 0 PandQ.
x2 + y2 + 2x − 3y + 6 = 0 x2 + y2 + x − 8y − 31 = 0? (1, − 2)(1, 4) (1, 2) 1, 4)
x2 + (y − 1)2 = 9, (x − 1)2 + y2
= 25.
aa.
120
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If the two circles andcutorthogonally,thenfindthevalueof .
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121
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Find the condition that the circle lies entirelywithin thecircle .
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122
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Findtheradicalcenterofthecircles
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Two circles and intersect in such a way that their common chord is of
2x2 + 2y2 − 3x + 6y + k = 0x2 + y2 − 4x + 10y + 16 = 0 k
(x− 3)2 + (y − 4)2 = r2
x2 + y2 = R2
x2 + y2 + 4x + 6y = 19, x2 + y2
= 9, x2 + y2 − 2x − 4y = 5,
C1 C2
123 maximumlength.Thecenterof is(1,2)anditsradiusis3units.Theradiusof
is5units.Iftheslopeofthecommonchordis thenfindthecenterof
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124
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
The equation of a circle is Find the center of the smallest circletouchingthecircleandtheline
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125
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Consider four circles . Find the equation of the smallercircletouchingthesefourcircles.
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126
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Find the equation of the circle whose radius is 3 and which touches internally thecircle atthepoint
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127
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Find the number of common tangents that can be drawn to the circlesand
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C1 C2
,34
C2 .
x2 + y2 = 4.x + y = 5√2
(x± 1)2 + (y ± 1)2 = 1
x2 + y2 − 4x − 6y = − 12 = 0 ( − 1, − 1).
x2 + y2 − 4x − 6y − 3 = 0 x2 + y2 + 2x + 2y + 1 = 0
128
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Two fixed circles with radii , respectively, touch each otherexternally.Thenidentifythelocusofthepointofintersectionoftheirdirectioncommontangents.
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129
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Two circles with radii touch each other externally such that is the anglebetween the direct common tangents, . Then prove that
.
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130
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If the radii of the circles andareincreasinguniformlyw.r.t. timeas0.3units/sand0.4
unit/s,respectively,thenatwhatvalueof willtheytoucheachother?
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131
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
If the circle cuts at,thenfindtheequationofthecircleon asdiameter.
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r1andr2, (r1 > r2)
aandb θ(a > b ≥ 2)
θ = 2 sin−1( )a − ba + b
(x − 1)2 + (y − 2)2 + (y − 2)2 = 1( − 7)2 + (y − 10)2 = 4
t
x2 + y2 + 2x + 3y + 1 = 0 x2 + y2 + 4x+ 3y + 2 = 0AandB AB
132
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Show that the equation of the circle passing through (1, 1) and the points ofintersection of the circles and
is
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133
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Find the equation of the smallest circle passing through the intersection of the lineandthecircle
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134
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
and arecircleofunitradiuswithcentersat(0,0)and(1,0),respectively, isacircleofunitradius.Itpassesthroughthecentersofthecircles andhasits center above the x-axis. Find the equation of the common tangent towhichdoesnotpassthrough
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135
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Findtheradiusof thesmallestcirclewhichtouchesthestraight line atandalsotouchestheline .Computeuptooneplaceofdecimalonly.
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136
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Avariablecirclewhichalwaystouchestheline at(1,1)cutsthecircle . Prove that all the common chords of intersection
passthroughafixedpoint.Findthatpoints.
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137
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Let be a circle passing through and and be a circle ofradius unitssuchthat isthecommonchordof Findtheequationof
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x2 + y2 + 13x − 13y = 02x2 + 2y2 + 4x − 7y − 25 = 0 4x2 + 4y2 + 30x − 13y − 25 = 0.
x + y = 1 x2 + y2 = 9
C1 C2 C3C1andC2
C1andC3C2.
3x− y = 6(1, − 3) y = x
x + y − 2 = 0x2 + y2 + 4x + 5y − 6 = 0
S1 A(0, 1) B( − 2, 2) S2√10 AB S1andS2.
S2 .
138
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
If belongtoafamilyofcirclesthroughthepointsprovethattheratioofthelengthofthetangentsfromanypointon tothecircles
isconstant.
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139
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Provethatquadrilateral ,where
isconcyclic.Alsofindtheequationofthecircumcircleof
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140
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Ifalinesegement movesintheplane remainingparallelto sothattheleftendpoint slidesalongthecircle thenthelocusof
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141
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
The tangents to having inclinations and intersect at If,thenfindthelocusof
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
C1, C2, andC3 (x1, y2)and(x2, y2)C1
C2andC3
ABCDAB ≡ x + y − 10,BC ≡ x − 7y+ 50 = 0, CD ≡ 22x − 4y + 125= 0, andDA ≡ 2x − 4y − 5 = 0,
ABCD.
AM = a XOY OXA x2 + y2 = a2, M .
x2 + y2 = a2 α β P .cotα + cot β = 0 P .
2 2 2
142 The locusof thepoint of intersectionof the tangents to the circle atpointswhoseparametricanglesdifferby .
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143
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Findthelocusofthecenterofthecircletouchingthecircleinternallyandtangentsonwhichfrom(1,2)aremakingof witheachother.
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144
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Tworodsoflengths slidealongthe and respectively,insuchamannerthattheirendsareconcyclic.Findthelocusofthecenterofthecirclepassingthroughtheendpoints.
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145
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
A circle with center at the origin and radius equal to a meets the axis of atand aretwopointsonthecirclesothat ,where is
aconstant.Findthelocusofthepointofintersectionof and
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146
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Findthelocusofthemidpointofthechordsofthecircle whichsubtendarightangleatthepoint
x2 + y2 = a2π
3
x2 + y2 − 4y − 2x = 4600
aandb x − y − aξs,
x
AandB.P(α) Q(β) α − β = 2y γ
AP BQ.
x2 + y2 = a2
(c, 0).
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147
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Avariablecirclepassesthroughthepoint andtouchesthex-axis.Showthatthelocusoftheotherendofthediameterthrough is
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148
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Find the locusof themidpoint of the chordof the circle ,whichmakesanangleof atthecenter.
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149
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
A tangent is drawn to each of the circles and Showthat if the two tangents are mutually perpendicular, the locus of their point ofintersectionisacircleconcentricwiththegivencircles.
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150
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
The circle is inscribed in a variable triangleSides and lie along the x- andy-axis, respectively,where is the origin.Findthelocusofthemidpointofside
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based On
A(a, b)A (x− a)2 = 4by.
x2 + y2 − 2x − 2y = 01200
x2 + y2 = a2 x2 + y2 = b2 .
x2 + y2 − 4x − 4y + 4 = 0 OAB.OA OB O
AB.
151
Locus
Apointmovessothatthesumofthesquaresoftheperpendicularsletfallfromitonthesidesofanequilateraltriangleisconstant.Provethatitslocusisacircle.
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152
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
isthevariablepointonthecirclewithcenterat and areperpendicularsfrom onthex-andthey-axis, respectively.Showthat the locusof thecentroidoftriangle isacirclewithcenteratthecentroidoftriangle andradiusequaltotheone-thirdoftheradiusofthegivencircle.
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153
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Tangents are drawn to the circle from two points on the axis ofequidistant from the point Show that the locus of their intersection is
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154
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Astraightlinemovessothattheproductofthelengthoftheperpendicularsonitfromtwofixedpointsisconstant.Provethatthelocusofthefeetoftheperpendicularsfromeachofthesepointsuponthestraightlineisauniquecircle.
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155
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Findthelocusofcenterofcircleofradius2units,ifinterceptcutonthex-axisistwiceofinterceptcutonthey-axisbythecircle.
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P C.CA CB
CPAB CAB
x2 + y2 = a2 x,(k, 0).
ky2 = a2(k − x).
156
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
ABCDisarectangle.AcirclepassingthroughvertexCtouchesthesidesABandADatMandN respectively. If thedistance lof the lineMN from thevertexC isPunitsthentheareaofrectangleABCDis
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157
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Let beatriangleright-angledat beitscircumcircle.Let bethecircletouchingthelines and andthecircle internally.Further,let bethecircletouchingthelines and producedandthecircle externally.If and aretheradiiofthecircles and ,respectively,showthat area
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158
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Therangeofparameter forwhichthevariableline liesbetweenthecircles and withoutintersecting or touching either circle is
(d)
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159
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
Findthelocusofthecentersofthecircles ,whereand are parameters, if the tangents from the origin to each of the circles areorthogonal.
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
ABC AandS S1AB AC S S2AB AC S r1 r2S1 S2 r1r2 = 4 (ABC).
' a' y = 2x+ ax2 + y2 − 2x − 2y + 1 = 0 x2 + y2 − 16x− 2y + 61 = 0
a ∈ (2√5 − 15, 0)a ∈ ( − ∞, 2√5 − 15, ) a ∈ (0, − √5 − 10) a ∈ ( −√5 − 1,∞)
x2 + y2 − 2ax− 2by + 2 = 0 ab
2 2
160(C) 2 45. Three concentric circles of which the biggest is , have theirradii inA.P If the line cutsall thecircles in realanddistinctpoints.TheintervalinwhichthecommondifferenceoftheA.Pwilllieis:
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161
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Let and beanylinepassingthrough(4,1)havingslope Find therangeof forwhichthereexist twopointson atwhichsubtendsarightangle.
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162
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Theequationof radicalaxisof twocircles is .Oneof thecircleshas theendsofadiameteratthepoints and theotherpasses throughthepoint(1,2).Findtheequatingofthesecircles.
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163
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
isacirclehavingthecenterat andradius`b(bWatchFreeVideoSolutiononDoubtnut
164
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
represents a circle. The equation gives two identicalsolutions: .Theequation giventwosolutions: Findtheequationofthecircle.
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x2 + y2 = 1y = x+ 1
A ≡ ( − 1, 0), B ≡ (3, 0), PQm. m PQ AB
x + y = 1(1, − 3) and (4, 1)
S (0, a)
S(x, y) = 0 S(x, 2) = 0x = 1 S(1, y) = 0 y = 0, 2.
165
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Findtheequationofthefamilyofcirclestouchingthelines
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166
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
are two points in the xy-plane, which are units distance apart andsubtendanangleof at thepoint on the line ,which islarger thananyanglesubtendedby the linesegment atanyotherpointon theline.Findtheequation(s)ofthecirclethroughthepoints
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167
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Fromthevariablepoint oncircle two tangentsaredrawn to thecircle which meet the curve at Find the locus of thecircumcenterof
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168
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Findthecircleofminimumradiuswhichpassesthroughthepoint(4,3)andtouchesthecircle externally.
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Two variable chords of a circle are such that
x2 − y2 + 2y − 1 = 0.
AandB 2√2900 C(1, 2) x − y + 1 = 0
ABA, BandC .
A x2 + y2 = 2a2,x2 + y2 = a2 BandC.
ABC .
x2 + y2 = 4
ABandBC x2 + y2 = r2
= =
169 .Findthelocusofthepointofintersectionoftangentsat
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170
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
If is a tangent to a circle whose center is , then find theequationoftheothertangenttothecirclefromtheorigin.
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171
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Findthelengthofthechordofcontactwithrespecttothepointonthedirectorcircleofcircle .
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172
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
Acircle is thedirectorcircleofcircle ,isthedirectorcircleofcircle ,andsoon.Ifthesumofradiiofallthesecirclesis2,thenfindthevalueofc.
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173
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Consider three circles such that is the director circle of is the director circlé of . Tangents to , from any point on
intersect ,at .Findtheanglebetweenthetangentsto atPandQ.Alsoidentifythelocusofthepointofintersec-tionoftangentsat .
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AB = BC = r AandC .
3x + y = 0 (2, − 1)
x2 + y2 + 2ax − 2by + a2 − b2 = 0
x2 + y2 + 4x − 2√2y + c = 0 S1 and S2S1
C1,C2 and C3 C2C1, and C3 C2 C1 C3
C2 P 2 and Q C22
P and Q
174
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
From a point on the normal of the circle two tangents are drawn to the same circle
touching it at point . If the area of quadrilateral (where is thecenterofthecircle)is36sq.units,findthepossiblevaluesof Itisgiventhatpointisatdistance fromthecircle.
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175
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Find the center of the smallest circle which cuts circles andorthogonally.
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176
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Perpendiculars are drawn, respectively, from the points to the chords ofcontact of the points with respect to a circle. Prove that the ratio of thelengthsofperpendiculars isequaltotheratioofthedistancesofthepointsfromthecenterofthecircles.
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177
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Thenumber of suchpoints ,where a is any integer, lying inside theregion bounded by the circles and
,is
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178
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
If eight distinct points can be found on the curve such that fromeachpoint two mutually perpendicular tangents can be drawn to the circle
thenfindthetangeof
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P y = x+ cx2 + y2 − 2x − 4y + 5 − λ2 − 0,
BandC OBPC Oλ. P
|λ|(√2 − 1)
x2 + y2 = 1x2 + y2 + 8x + 8y − 33 = 0
PandQQandP
PandQ
(a + 1, √3a)x2 + y2 − 2x − 3 = 0
x2 + y2 − 2x − 15 = 0
|x| + |y| = 1
x2 + y2 = a2, a.
179
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Acircleofradius5unitshasdiameteralongtheanglebisectorofthelinesand If thechordofcontact fromtheoriginmakesanangleof withthepositivedirectionofthex-axis,findtheequationofthecircle.
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180
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Let bechordofcontactofthepoint w.r.tthecircle .Thenfindthelocusoftheorthocentreofthetriangle ,where isanypointmovingonthecircle.
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181
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
Let be anymoving point on the circle be the chord ofcontact of this point w.r.t. the circle . The locus of thecircumcenter of triangle being the center of the circle) is
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182
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
AB is a diameter of a circle. CD is a chord parallel to AB and . ThetangentatBmeetsthelineACproducedatEthenAEisequalto-
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
x + y = 2x − y = 2. 450
AB (5, − 5) x2 + y2 = 5PAB P
P x2 + y2 − 2x = 1. ABx2 + y2 − 2x = 0
CAB(C2x2 + 2y2 − 4x + 1 = 0 x2 + y2 − 4x + 2 = 0 x2 + y2 − 4x + 1 = 02x2 + 2y2 − 4x + 3 = 0
2CD = AB
183Twoparalleltangentstoagivencirclearecutbyathirdtangentatthepoint .Show that the lines from to the center of the circle are mutuallyperpendicular.
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184
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Acircleofradius1unittouchesthepositivex-axisandthepositivey-axisat ,respectively. A variable line passing through the origin intersects the circle at twopoints .Iftheareaoftriangle ismaximumwhentheslopeofthelineis
thenfindthevalueof
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185
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
The number of rational point(s) [a point (a, b) is called rational, if both arerationalnumbers]onthecircumferenceofacirclehavingcenter isatmostone(b)atleasttwoexactlytwo(d)infinite
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186
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
If the equation of any two diagonals of a regular pentagon belongs to the family oflines andtheir lengthsaresin , thenthelocusofthecenterofcirclecircumscribingthegivenpentagon(thetrianglesformedbythesediagonalswiththesidesofpentagonhavenosidecommon)is
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RandQRandQ
AandB
DandE DEBm, m−2
aandb(π, e)
(1 + 2λ)λ − (2 + λ)x + 1 − λ = 0 360
x2 + y2 − 2x − 2y + 1 + sin2 720
= 0x2 + y2 − 2x − 2y + cos2 720 = 0x2 + y2 − 2x − 2y + 1 + cos2 720
= 0x2 + y2 − 2x − 2y + sin2 720 = 0
187
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If are equal perpendicular chords of the circles , then the equations of are,where is the
origin. and and and and
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188
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Theequationofthechordofthecircle ,whichpassesthroughtheoriginsuchthattheorigindividesitintheratio4:1,is
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189
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Theline istangenttothecircleatthepoint(2,5)andthecenterofthecirclelieson Theradiusofthecircleis (b) (c) (d)
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190
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Ina triangle rightangledat on the leg asdiameter,a semicircle isdescribed. Ifachord joins with thepointof intersection of thehypotenuseand
thesemicircle, then the lengthof isequal to (b)
(d)
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191
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
A rhombus is inscribed in the region common to the two circles and with two of its vertices on
thelinejoiningthecentersofthecircles.Theareoftherhombusis (b) (d)noneofthese
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OAandOBx2 + y2 − 2x + 4y = 0 OAandOB O
3x + y = 0 3x − y = 0 3x + y = 0 3y − x = 0 x + 3y = 0y − 3x = 0 x + y = 0 x − y = 0
x2 + y2 − 3x − 4y − 4 = 0
2x − y + 1 = 0x − 2y = 4. 3√5 5√3 2√5
5√2
ABC, A, ACA D
ACAB
.AD
√AB2 + AD2
AB.AD
AB + AD
√AB .AD
AB.AD
√AB2 − AD2
x2 + y2 − 4x − 12 = 0 x2 + y2 + 4x − 12 = 08√3sq
.units
4√3sq.units 6√3sq
.units
192
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Thelocusfothecenterof thecirclessuchthatthepoint(2,3) isthemidpointof thechord is (a) (b) (c)
(d)noneofthese
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193
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Twocongruentcircleswithcenteredat(2,3)and(5,6)whichintersectatrightangles,haveradiusequalto2 (b)3(c)4(d)noneofthese
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194
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Acircleofradiusunityiscenteredatthetorigin.Twoparticlestartmovingatthesametimefromthepoint andmovearoundthecircleinoppositedirection.Oneoftheparticlemoves anticlockwisewith constant speed and the othermoves clockwisewithconstantspeed .Afterleaving ,thetwoparticlesmeetfirstatapointandcontinueuntiltheymeetnextatpoint .Thecoordinatesofthepoint are
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195
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Thevalueof'c'forwhichtheset
containsonlyonepointincommonis
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
5x + 2y = 16 2x − 5y + 11 = 0 2x + 5y − 11 = 02x+ 5y + 11 = 0
√3
(1, 0)v
3v (1, 0) P ,Q Q
{(x, y) ∣ x2 + y2 + 2x ≤ 1}
∩ {(x, y) ∣ x − y + c ≤ 0}
196
Acircleisinscribedti.e.touchesallfoursides)intoarhombousABCDwithoneangle60°.Thedistancefromthecentreofthecircletothenearestvertexisequalto1.IfPisanypointofthecirclethen isequalto:
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197
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Two circles with radii touch each other externally such that is the anglebetween the direct common tangents, . Then prove that
.
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198
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
arefixedpointshavingcoordinates(3,0)and respectively.Iftheverticalangle is ,thenthelocusofthecentroidof hasequation.(a)
(b) (c) (d)
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199
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
is a square of unit area. A circle is tangent to two sides of andpasses through exactly one of its vertices. The radius of the circle is (b)
(d)
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
|PA|2 + |PB|2 + |PC|2 + |PD|2
aandb θ(a > b ≥ 2)
θ = 2 sin−1( )a − ba + b
BandC ( − 3, 0),BAC 900 ABC
x2 + y2 = 1 x2 + y2 = 2 9(x2 + y2) = 1 9(x2 + y2) = 4
ABCD ABCD2 − √2
√2 − 1 1/21
√2
200
Apairoftangentsisdrawntoaunitcirclewithcenterattheoriginandthesetangentsintersectat enclosinganangleof .Theareaenclosedbythesetangentsand
thearcofthecircleis (b) (d)
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201
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
A straight line with slope 2 and y-intercept 5 touches the circle at a point Then the coordinates of are
(b) (d)
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202
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
A circle of constant radius passes through the origin and cuts the axes ofcoordinates at points and . Then the equation of the locus of the foot of
perpendicular from to is
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203
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
A linemeets the coordinate axes at and . A circle is circumscribed about thetriangle If aredistancesof thetangentstothecircleat theorigin
fromthepoints ,respectively,thenthediameterofthecircleis (b)
(d)
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A 600
−2
√3
π
6√3 −
π
3−
π
3√36
√3(1 − )π6
x2 + y2 + 16x+ 12y + c = 0 Q. Q( − 6, 11) ( − 9, − 13) ( − 10, − 15) ( − 6, − 7)
a OP Q
O PQ (x2 + y2)( + ) = 4a21
x21y2
(x2 + y2)2( + ) = a2
1
x21y2
(x2 + y2)2( + ) = 4a2
1
x21y2
(x2 + y2)( + ) = a21
x21y2
A BOAB. d1andd2 O
AandB2d1 + d2
2d1 + 2d2
2d1 + d2
d1d2
d1 + d2
204
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Ifacircleofconstantradius passesthroughtheorigin andmeetsthecoordinateaxes at , then the locus of the centroud of triangle is (a)
(b) (c) (d)
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205
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
Theequationofthelineinclinedatanangleof tothe suchthatthetwo
circles and interceptequallengthonit,is (b) (d)
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206
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Let be a circle with two diameters intersecting at an angle of A circle istangenttoboththediametersandto andhasradiusunity.Thelargestradiusofis (b) (d)noneofthese
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207
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
A straight line with equation meets the circle with equationat inthefirstquadrant.Alinethrough perpendicular to cuts
they-axisat .Thevalueof is12(b)15(c)20(d)25
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208
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Let representthelengthsofarighttriangleslegs.If isthediameterofacircleinscribed into the triangle, and is the diameter of a circle circumscribed on the
3k OAandB OAB
x2 + y2 = (2k)2 x2 + y2 = (3k)2 x2 + y2 = (4k)2 x2 + y2 = (6k)2
π
4X − aξs,
x2 + y2 = 4 x2 + y210x − 14y + 65 = 02x − 2y − 3 = 0 2x − 2y + 3 = 0 x − y + 6 = 0 x − y − 6 = 0
C 300 . SC C
1 + √6 + √2 1 + √6 −√2 √6 + √2 − 11
l1 x − 2y + 10 = 0x2 + y2 = 100 B B l1
P(o, t) t
aandb dD
1
triangle,the equals.(a) (b) (c) (d)
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209
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Ifthechord ofthecircles subtendsanangleof atthemajorsegmentofthecircle,thenthevalueof is (b) (c) (d)noneofthese
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210
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
Avariablechordofthecircle isdrawnfromthepoint meetingthe circle at the point and A point is taken on the chord such that
. The locus of is
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211
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
In triangle , the equation of side is The circumcenter andorthocentre of triangle are (2, 3) and (5, 8), respectively. The equation of thecircumcirle of the triangle is
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212
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
The range of values of r for which the point is an
interior point of themajor segment of the circle , cut-off by the line
d + D a + b 2(a + b) (a + b)12
√a2 + b2
y = mx + 1 x2 + y2 = 1 450m 2 −2 −1
x2 + y2 = 4 P (3, 5)A B. Q
2PQ = PA + PB Q x2 + y2 + 3x + 4y = 0 x2 + y2 = 36x2 + y2 = 16 x2 + y2 − 3x− 5y = 0
ABC BC x − y = 0.
x2 + y2 − 4x + 6y − 27 = 0x2 + y2 − 4x − 6y − 27 = 0 x2 + y2 + 4x+ 6y − 27 = 0x2 + y2 + 4x + 6y − 27 = 0
( − 5 + , − 3 + )r
√2
r
√2x2 + y2 = 16
+ = 2
,is:
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213
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
A square is inscribed in the circle with its sidesparalleltothecoordinateaxes.Thecoordinatesofitsverticesare
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214
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
aretheverticesofa .Theincircleofthetrianglehas equation.
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215
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If is the origin and are the tangents from the origin to the circle , thenthecircumcenterof triangle is
(b) (d)
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x + y = 2
x2 + y2 − 2x + 4y − 93 = 0
( − 6, − 9), ( − 6, 5), (8, − 9),
(8, 5)( − 6, − 9), ( − 6, − 5), (8, − 9),
(8, 5)( − 6, − 9), ( − 6, 5), (8, 9), (8, 5)( − 6, − 9), ( − 6, 5), (8, − 9),
(8, − 5)
( − 6, 0), 0, 6), and( − 7, 7) ABC
x2 + y2 − 9x− 9y + 36 = 0 x2 + y2 + 9x − 9y + 36 = 0x2 + y2 + 9x + 9y − 36 = 0 x2 + y2 + 18x − 18y + 36 = 0
O OPandOQx2 + y2 − 6x + 4y + 8 − 0 OPQ (3, − 2)
( , − 1)32
( , − )34
12
( − , 1)32
216
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Thelocusofthemidpointofalinesegmentthatisdrawnfromagivenexternalpointtoagivencirclewithcenter (where istheorgin)andradius isastraightline
perpendiculat to acirclewithcenter andradius acirclewithcenter andradius acirclewithcenteratthemidpoint andradius
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217
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Thedifferencebetween theradiiof the largestandsmallestcircleswhichhave theircentresonthecircumferenceofthecircle andpassesthroughpoint(a,b)lyingoutsidethecircleis:
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218
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Anisoscelestriangle isinscribedinacircle withthevertex at and the base angle each equal . Then the coordinates of an
endpoint of the base are. (b) (d)
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219
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Theequationof four circlesare .The radiusofacircletouchingallthefourcirclesis (b) (d)
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220
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thelocusofapointwhichmovessuchthatthesumofthesquareofitsdistancefromthreeverticesofatriangleisconstantisa/ancircle(b)straightline(c)ellipse(d)noneofthese
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
P O O rPO P r P
2r PO .r
2
x2 + y2 + 2x + 4y − 4 = 0
ABC x2 + y2 = a2 A
(a, 0) BandC 750
( − , )√3a2
a
2( − , a)√3a
2( , )a
2√3a2
( , − )√3a2
a
2
(x ± a)2 + (y ± a2 = a2
(√2 + 2)a 2√2a (√2 + 1)a (2 + √2)a
221Acircle passes through thepoints and touches the y-axis at
If is maximum, then (a) (b) (c) (d)
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222
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Acirclewithcenter passes through theorigin.Theequationof the tangent tothe circle at the origin is (b) (d)
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223
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Theareaofthetriangleformedbyjoiningtheorigintothepointofintersectionoftheline andthecircle is3(b)4(c)5(d)6
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224
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If isapointonthecirclewhosecenterisonthex-axisandwhichtouchestheline at then the greatest value of is (a) (b) 6 (c)
(d)
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225
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Aregion in the plane isboundedby thecurve and the line.Ifthepoint liesintheinterioroftheregion,then (b)
(d)noneofthese
A(1, 0)andB(5, 0),C(0, h). ∠ACB h = 3√5 h = 2√5 h = √5h = 2√10
(a, b)ax − by = 0 ax + by = 0 bx − ay = 0
bx + ay = 0
x√5 + 2y = 3√5 x2 + y ∘ = 10
(α, β)x + y = 0 (2, − 2), α 4 − √2
4 + 2√2 +√2
x− y y = √25 − x2
y = 0 (a, a + 1) a ∈ ( − 4, 3)a ∈ ( −∞, − 1) ∈ (3, ∞) a ∈ ( − 1, 3)
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226
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Therearetwocircleswhoseequationare and
If the two circles have exactly two common tangents, then the number of possiblevaluesof is2(b)8(c)9(d)noneofthese
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227
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
isacircleofradius1touchingthex-andthey-axis. isanothercircleofradiusgreaterthan1andtouchingtheaxesaswellasthecircle .Thentheradiusofis (b) (d)noneofthese
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228
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
The equation of the incircle of equilateral triangle where and lies in the fourth quadrant is
noneofthese
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
x2 + y2 = 9x2 + y2 − 8x − 6y + n2 = 0, n ∈ Z.
n
C1 C2C1 C2
3 − 2√2 3 + 2√2 3 + 2√3
ABCB ≡ (2, 0),C ≡ (4, 0), A
x2 + y2 − 6x + + 9 = 02y
√3x2 + y2 − 6x − + 9 = 0
2y
√3
x2 + y2 + 6x + + 9 = 02y
√3
f(x, y) = x2 + y2 + 2ax + 2by + c= 0
( , 0) = 0 2, (0, ) = 0
229representsacircle.If hasequalroots,eachbeing and
has 2 and 3 as its roots, then the center of the circle is (b)Data are not
sufficient (d)Dataareinconsistent
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230
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Theareaboundedbythecircles andthepairof lines
isequalto (b) (c) (d)
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231
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
The straight line will touch the circle if(b) (d)noneofthese
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232
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Let be a quadrilateral with are , side parallel to the side . Let be perpendicular to . If a circle is
drawninsidethequadrilateral touchingallthesides,thenitsradiusis (b)2
(c) (d)1
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233
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Consider a family of circleswhich are passing through the point and aretangenttothex-axis.If arethecoordinatesofthecenterofthecircles,thenthe
setofvaluesof isgivenbytheinterval. (b) (d)`0
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f(x, 0) = 0 2, f(0, y) = 0
(2, )52
( − 2, − )52
x2 + y2 = 1, x2 + y2 = 4,
√3(x2 + y2) = 4xyπ
25π2
3ππ
4
x cos θ + y sin θ = 2 x2 + y2 − 2x = 0θ = nπ, n ∈ IQ A = (2n + 1)π, n ∈ I θ = 2nπ, n ∈ I
ABCD 18 ABCD, andAB = 2CD AD ABandCD
ABCD 332
( − 1, 1)(h, k)
k k ≥12
− ≤ k ≤12
12k ≤
12
234
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Iftheconicswhoseequationsare
intersect at four concyclic points, then, (where (b) (d)noneofthese
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235
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Therangeofvaluesof suchthattheangle betweenthepairof tangents
drawnfrom tothecircle liesin is(a)
(b) (c) (d)noneofthese
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236
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Thecirclewhichcanbedrawntopassthrough(1,0)and(3,0)andtotouchthey-axis
intersectatangle Then isequalto (b) (c) (d)
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237
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Thelocusofthemidpointsofthechordsofcontactof fromthepointsontheline isacirclewithcenter If istheorigin,then isequalto
2(b)3(c) (d)
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S ≡ sin2 θx2 + 2hxy + cos2 θy2
+ 32x + 16y + 19 = 0, S ′
≡ cos2 θx2 + 2h ′xy + s ∈2 θy2
+ 16x + 32y + 19 = 0θ ∈ R) h + h ′ = 0 h = h'
h + h ′ = 1
λ, λ > 0 θ
(λ, 0) x2 + y2 = 4 ( , )π
22π3
( , )4
√3
2
√2(0, √2) (1, 2)
θ. cos θ12
−12
14
−14
x2 + y2 = 23x + 4y = 10 P . O OP
12
13
238
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Suppose , where are in be normal to a family ofcircles. The equation of the circle of the family intersecting the circle
orthogonally is (a) (b) (c) (d)
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239
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Ifacircleofradius istouchingthelines inthefirstquadrantat
points ,thentheareaoftriangle beingtheorigin)is(a) (b)
(c) (d)
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240
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Let be anymoving point on the circle be the chord ofcontact of this point w.r.t. the circle . The locus of thecircumcenter of triangle being the center of the circle) is
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ax+ by + c = 0 a, bandc AP
x2 + y2 − 4x − 4y − 1 = 0 x2 + y2 − 2x + 4y − 3 = 0x2 + y2 − 2x + 4y + 3 = 0 x2 + y2 + 2x + 4y + 3 = 0x2 + y2 + 2x − 4y + 3 = 0
r x2 − 4xy + y2 = 0
AandB OAB(O 3√3r2
4√3r2
43r2
4r2
P x2 + y2 − 2x = 1. ABx2 + y2 − 2x = 0
CAB(C2x2 + 2y2 − 4x + 1 = 0 x2 + y2 − 4x + 2 = 0 x2 + y2 − 4x + 1 = 02x2 + 2y2 − 4x + 3 = 0
241
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Twocirclesof radii touchingeachotherexternally,are inscribed in thearea
boundedby andthex-axis.If then isequalto (b) (c)
(d)
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242
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Fromanarbitrarypoint onthecircle ,tangentsaredrawntothecircle , which meet at . The locus of the point of
intersectionoftangentsat tothecircle is
(b) (d)noneofthese
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243
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If are the radii of the smallest and the largest circles, respectively, which
passthough(5,6)andtouchthecircle then is (b)
(d)
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244
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
The minimum radius of the circle which is orthogonal with both the circlesand is4(b)3(c) (d)1
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245
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If isacircleand and areapairoftangentsonwhere isanypointonthedirectorcircleof thentheradiusofthesmallest
circlewhichtouches externallyandalsothetwotangents and is(b) (d)1
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aandb
y = √1 − x2 b = ,12
a14
18
12
1
√2
P x2 + y2 = 9x2 + y2 = 1 x2 + y2 = 9 AandB
AandB x2 + y2 = 9 x2 + y2 = ( )2277
x2 − y2( )2277
y2 − x2 = ( )2277
r1andr2
(x − 2)2 + y2 = 4, r1r2441
414
541
416
x2 + y2 − 12x+ 35 = 0 x2 + y2 + 4x+ 3 = 0 √15
C1 : x2 + y2 = (3 + 2√2)2
PA PB
C1, P C1,c1 PA PB 2√3 − 3
2√2 − 1 2√2 − 1
246
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Fromapoint ,twotangents aredrawntoagivencirclewhoseradiusis5.Ifthecircumcenteroftriangle is(2,3),thentheequationofthe circle is
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247
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
If the tangents are drawn from any point on the line to the circle,thenthechordofcontactpassesthroughthepoint.(3,5)(b)(3,3)(c)
(5,3)(d)noneofthese
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248
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
If the radius of the circumcircle of the triangle TPQ, where PQ is chord of contactcorresponding topointTwith respect to circle , is 6units,thenminimumdistancesofTfromthedirectorcircleofthegivencircleis
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249
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
isapoint in thefirstquadrant. If thetwocircleswhichpassthrough andtouch both the coordinates axes cut at right angles, then
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases Of
R(5, 8) RPandRQ S = 0PQR
S = 0 x2 + y2 + 2x + 4y − 20 = 0 x2 + y2 + x + 2y − 10 = 0x2 + y2 − x+ 2y − 20 = 0 x2 + y2 + 4x − 6y − 12 = 0
x + y = 3x2 + y2 = 9
x2 + y2 − 2x + 4y − 11 = 0
P (a, b) P
a2 − 6ab + b2 = 0a2 + 2ab − b2 = 0 a2 − 4ab + b2 = 0 a2 − 8ab + b2 = 0
250
TwoCircles
Thenumberofcommontangent(s)tothecircles andis1(b)2(c)3(d)4
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251
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Two circles of radii 4cm and 1cm touch each other externally and is the angle
containedbytheirdirectcommontangents.Then isequalto (b) (d)
noneofthese
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252
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thelocusofthemidpointsofthechordsofthecircle which
subtend a right angle at is
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253
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Fromthepoints(3,4),chordsaredrawntothecircle .Thelocusof the midpoints of the chords is (a) (b)
(c) (d)
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
x2 + y2 + 2x + 8y − 23 = 0x2 + y2 − 4x − 10y − 19 = 0
θ
sin θ2425
1225
34
x2 + y2 − ax − by = 0
( , )a
2b
2ax + by = 0 ax + by = a2 = b2
x2 + y2 − ax − by + = 0a2 + b2
8x2 + y2 − ax− by − = 0
a2 + b2
8
x2 + y2 − 4x = 0x2 + y2 − 5x − 4y + 6 = 0
x2 + y2 + 5x − 4y + 6 = 0 x2 + y2 − 5x + 4y + 6 = 0x2 + y2 − 5x − 4y − 6 = 0
254
Theanglesatwhichthecircles
intersectis (b) (c) (d)
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255
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Atangentatapointonthecircle intersectsaconcentriccircle attwopoints .Thetangentstothecircle at meetatapointonthecircle
Then the equation of the circle is
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256
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Tangent are drawn to the circle at the points where it is met by thecircles
being the variable. The locus of the point of intersection of these tangents is(b) (d)
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257
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
If points are (1, 0) and (0, 1), respectively, and point is on the circle , then the locusof theorthocentreof triangle is
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(x − 1)2 + y2 = 10andx2
+ (y − 2)2 = 5π
6π
4π
3π
2
x2 + y2 = a2 CPandQ X PandQ
x2 + y2 = b2 . x2 + y2 = abx2 + y2 = (a − b)2 x2 + y2 = (a + b)2 x2 + y2 = a2 + b2
x2 + y2 = 1
x2 + y2 − (λ + 6)x + (8 − 2λ)y− 3 = 0, λ
2x− y + 10 = 0 2x + y − 10 = 0 x − 2y + 10 = 0 2x + y − 10 = 0
AandB Cx2 + y2 = 1 ABC x2 + y2 = 4x2 + y2 − x− y = 0 x2 + y2 − 2x − 2y + 1 = 0x2 + y2 + 2x − 2y + 1 = 0
258
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If the line is theequationofa transversecommon tangent tothecircles and ,thenthevalueof
is (b) (c) (d)
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259
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Any circle through the point of intersection of the lines and intersects these linesatpoints .Then theanglesubtended
bythearc atitscenteris (b) (c) dependsoncenterandradius
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260
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
Iftheanglebetweenthetangentsdrawnto from(0,0)is then
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261
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Thecommonchordof thecircle andacirclepassingthrough the origin and touching the line always passes through the point.
(b)(1,1) (d)noneofthese
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262
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
The chords of contact of tangents from three points to the circleareconcurrent.Then will (a)beconcyclic (b)becollinear
(c)formtheverticesofatriangle(d)noneofthese
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263
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
The chord of contact of tangents from a point to a circle passes through Ifarethelengthofthetangentsfrom tothecircle,then isequalto
(b) (d)
x cos θ + y sin θ = 2x2 + y2 = 4 x2 + y2 − 6√3x − 6y + 20 = 0 θ
5π6
2π3
π
3π
6
x +√3y = 1√3x − y = 2 PandQ
PQ 1800 900 1200
x2 + y2 + 2gx + 2fy + c = 0,
π
2g2 + f 2 = 3c g2 + f2 = 2c g 2 + f2 = 5c g2 + f 2 = 4c
x2 + y2 + 6x + 8y − 7 = 0y = x
( − , )12
12
( , )12
12
A, BandCx2 + y2 = a2 A, BandC
P Q.l1andl2 PandQ PQl1 + l2
2l1 − l2
2√l12 + l22 2√l12 + l22
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264
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Let are circles defined by and. The length of the shortest line segment PQ that is
tangentto atPandto atQis
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265
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Theendsofaquadrantofacirclehave thecoordinates (1,3)and (3,1).Then thecenterofsuchacircleis
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266
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
If the line is a normal to the circle and atangenttothecircle ,then
WatchFreeVideoSolutiononDoubtnut
267
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Theradiusofthetangentcirclethatcanbedrawntopassthroughthepoint(0,1)and(0,6)andtouchingthex-axisis(a)5/2(b)Â3/2(c)7/2(d)9/2
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C1 and C2 x2 + y2 − 20x + 64 = 0x2 + y2 + 30x+ 144 = 0
C1 C2
ax + by = 2 x2 + y2 − 4x− 4y = 0x2 + y2 = 1 a and bare
268
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Theequationof the tangent to thecircle whichmakesa triangleofarea with the coordinate axes, is (b)
(d)
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269
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
If the circle is touched by at such thatthenthevalueof is36(b)144(c)72(d)noneofthese
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270
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
The number of points lying inside or on the circle andsatisfyingtheequation is______
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271
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Theareaofthetriangleformedbythepositive axisandthenormalandtangentto the circle at is (a) (b) (c)
(d)noneofthese
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
x2 + y2 = a2,a2 x± y = a√2 x± y = ± a√2
x ± y = 2a x + y = ± 2a
x2 + y2 + 2gx + 2fy + c = 0 y = x POP = 6√2, c
P(x, y) x2 + y2 = 9tan4 x+ cot4 x + 2 = 4 sin2 y
x−x2 + y2 = 4 (1, √3) 2√3sq
.units 3√2sq
.units
√6sq.units
272
Let beapointonthecircle apointontheline ,and the perpendicular bisector of be the line . Then the
coordinatesof are (b) (d)
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273
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Astraight linemoves such that thealgebraic sumof theperpendicularsdrawn to itfromtwofixedpointsisequalto .Then, thenstraight linealwaystouchesafixed
circleofradius. (b) (c) (d)noneofthese
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274
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thecoordinatesofthemiddlepointofthechordcut-offby bythecircle are (b) (c) (d)
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275
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
The locus of the point from which the lengths of the tangents to the circles and are equal is a straight line
inclinedat withthelinejoiningthecentersofthecirclesacircle(c)anellipse(d)a
straightlineperpendiculartothelinejoiningthecentersofthecircles.
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
The equation of the circumcircle of an equilateral triangle is
P x2 + y2 = 9, Q 7x + y + 3 = 0PQ x − y + 1 = 0
P (0, − 3) (0, 3) ( , )7225
2135
( − , )7225
2125
2k
2kk
2k
2x − 5y + 18 = 0x2 + y2 − 6x + 2y − 54 = 0 (1, 4) (2, 4) (4, 1) (1, 1)
x2 + y2 = 4 2(x2 + y2) − 10x+ 3y − 2 = 0π
4
2 2
276
and one vertex of the triangle in (1, 1). Theequationoftheincircleofthetriangleis
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277
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Tangents and aredrawnto fromanyarbitrarypoint ontheline . The locus of the midpoint of chord is
WatchFreeVideoSolutiononDoubtnut
278
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Thecircleshaving radii intersectorthogonally.The lengthof theircommonchordis`
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279
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
If thepairofstraight lines is tangent to thecircleat fromthe origin such that the area of the smaller sector formed by is
where isthecenterofthecircle,the equals (b) (c)
3(d)
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280
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
The two circles which pass through and touch the line will intersect each other at right angle if
(d)
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281
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Theconditionthatthechord of maysubtendarightangleatthecenterofthecircleis
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x2 + y2 + 2gx + 2fy + c = 04(x2 + y2) = g2 + f2
4(x2 + y2) = 8gx+ 8fy
= (1 − g)(1 + 3g)
+ (1 − f)(1 + 3f)4(x2 + y2) = 8gx+ 8fy = g2 + f2 noneofthese
PA PB x2 + y2 = 9 Px + y = 25 AB
25(x2 + y2) = 9(x + y) 25(x2 + y2) = 3(x+ y) 5(x2 + y2) = 3(x + y)noneofthese
r1andr2
xy√3 − x2 = 0 PandQO CPandCQ
3πsq.unit, C OP
(3√3)2
3√3
√3
(0, a)and(0, − a)y = mx + c a2 = c2(2m + 1)a2 = c2(2 +m2) c2 = a2(2 + m2) c2 = a2(2m + 1)
x cosα + y sinα − p = 0 x2 + y2 − a2 = 0
282
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Thelocusofthemidpointofachordofthecircle whichsubtendsarightangleattheoriginsis (b) (d)
WatchFreeVideoSolutiononDoubtnut
283
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
Ifthechordofcontactoftangentsfromapoint toagivencirclepassesthroughthen the circle on as diameter. cuts the given circle orthogonally touches thegivencircleexternallytouchesthegivencircleinternallynoneofthese
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284
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Letthebase ofatriangle befixedandthevertex liesonafixedcircleofradius Linesthrough aredrawntointersect respectively,at
suchthat .Ifthepointofintersection of these lines lies on themedian through for all positions of then the
locusof isacircleofradius acircleofradius aparabolaoflatusrectum a
rectangularhyperbola
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285
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of TheChordBisectedAtAGivenPoints
Thenumberof intergralvalueof forwhich thechordof thecirclepassingthroughthepoint getsbisectedatthepoint andhasintegralslopeis8(b)6(c)4(d)2
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x2 + y2 = 4x + y = 2 x2 + y2 = 1 x2 + y2 = 2 x + y = 1
P Q,PQ
AB ABC Cr. AandB CBandCA,
EandF CE :EB = 1: 2andCF :FA = 1: 2P AB AB,
Pr
22r 4r
y x2 + y2 = 125P(8, y) P(8, y)
286
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Considersquare ofsidelength1.Let bethesetofallsegmentsoflength1withendpointsontheadjacentsidesofsquare .Themidpointsofsegmentsin enclosearegionwitharea Thevalueof is(a) (b) (c) (d)
noneofthese
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287
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
The rangeofvaluesof forwhich the line isanormal to thecircle for all values of is (a) (b) (c)
(d)
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288
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
)Sixpoints(x,yi),i=1,2, ,.., 6are takenon thecirclex4such that thecirclex2+y4such that 6 6 J:1-8 and Σ, 4 . The line segment X,=8 and and 2-yi = 4. The linesegment x1 i=1 joining orthocentre of a triangle formed by any three points andcentroidofa triangle formedbyother threepointspasses througha fixed i=1points(h,k),thenh+kisA)1B)2C)3D)4
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289
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
A circle with radius and center on the y-axis slied along it and a variable linethrough (a, 0) cuts the circle at points . The region in which the point ofintersection of the tangents to the circle at points and lies is represented by
(b) (d)
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
ABCD PABCD
P A. Aπ
41 −
π
44 −
π
4
α 2y = gx + αx2 = y2 + 2gx + 2gy − 2 = 0 g [1,∞) [ − 1, ∞)(0, 1) ( − ∞, 1]
|a|PandQ
P Qy2 ≥ 4(ax − a2) y2 ≤ 4(ax − a2) y ≥ 4(ax − a2) y ≤ 4(ax − a2)
290If the angle of intersection of the circle and
is ,thentheequationofthelinepassingthrough(1,2)andmaking an angle with the y-axis is (a) (b) (c) (d)
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291
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Thecentersofasetofcircles,eachofradius3,lieonthecircle .Thelocusofanypointinthesetis (d)
WatchFreeVideoSolutiononDoubtnut
292
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Thecoordinatesoftwopoints are istheorigin.Ifthe circles are described on as diameters, then the length of their
common chord is (b) (d)
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293
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Consideracircle lyingcompletelyinthefirstquadrant.
If arethemaximumandminimumvaluesof forallorderedpairs
onthecircumferenceof thecircle, thenthevalueof is(a) (b)
(c) (d)
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x2 + y2 + x + y = 0x2 + y2 + x− y = 0 θ
θ x = 1 y = 2 x+ y = 3x − y = 3
x2 + y2 = 254 ≤ x2 + y2 ≤ 64 x2 + y2 ≤ 25 x2 + y2 ≥ 25
3 ≤ x2 + y2 ≤ 9
PandQ (x1, y1)and(x2, y2)andOOPandOQ
|x1y2 + x2y1|PQ
|x1y2 − x2y1|PQ
|x1x2 + y1y2|PQ
|x1x2 − y1y2|PQ
x2 + y2 + ax + by + c = 0m1andm2
y
x(x, y)
(m1 + m2)a2 − 4c
b2 − 4c2abb2 − 4c
2ab4c − b2
2abb2 − 4ac
294
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
The equation of the circle passing through the point of intersection of the circle and the line and havingminimumpossible radius is (a)
(b) (c)(c)
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295
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
The locus of the center of the circle touching the line at is (a)(b) (c) (d)noneofthese
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296
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Thedistancefromthecenterofthecircle tothecommonchordofthecircles and is2(b)4
(c) (d)
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297
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
The equation of the circle passing through the point of intersection of the circles and and the point is
(a) (b) (c)(d)
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
x2 + y2 = 4 2x+ y = 15x2 + 5y2 + 18x+ 6y − 5 = 0 5x2 + 5y2 + 9x+ 8y − 15 = 05x2 + 5y2 + 4x + 9y − 5 = 0 5x2 + 5y2 − 4x − 2y − 18 = 0
2x− y = 1 (1, 1)x + 3y = 2 x + 2y = 2 x + y = 2
x2 + y2 = 2xx2 + y2 + 5x − 8y + 1 = 0 x2 + y2 − 3x + 7y − 25 = 0
3413
2617
x2 + y2 − 4x − 2y = 8 x2 + y2 − 2x − 4y = 8 ( − 1, 4)x2 + y2 + 4x + 4y − 8 = 0 x2 + y2 − 3x + 4y + 8 = 0
x2 + y2 + x+ y = 0 x2 + y2 − 3x − 3y − 8 = 0
2 2 2 2
298 If the radiiof thecircle andareincreasinguniformlyw.r.t.timesas0.3unit/sisand0.4unit/s,thentheywilltoucheachotherat equalto (b) (c) (d)
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299
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
The equation of the circlewhich has normals and a tangent is
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300
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Awheelofradius8unitsrollsalongthediameterofasemicircleofradius25units;itbumps into this semicircle. What is the length of the portion of the diameter thatcannotbetouchedbythewheel?12(b)15(c)17(d)20
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301
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Thepoint([p+1],[p])islyinginsidethecircle .Thenthesetofallvaluesof is(where[.]representsthegreatestintegerfunction)(a) (b)
(c) (d)
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302
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Thesquared lengthof the interceptmadeby the line on thepairof tangentsdrawn from the origin to the circle is
(x − 1)2 + (y − 2)2 = 1 (x − 7)2 + (y − 10)2 = 4
t 45s 90s 11s 135s
x − 1). (y − 2) = 03x+ 4y = 6 x2 + y2 − 2x − 4y + 4 = 0 x2 + y2 − 2x − 4y + 5 = 0x2 + y2 = 5 (x − 3)2 + (y − 4)2 = 5
x2 + y2 − 2x − 15 = 0p [ − 2, 3)
( − 2, 3) [ − 2, 0) ∪ (0, 3) [0, 3)
x = hx2 + y2 + 2gx + 2fy + c = 0
4 2 4 2 4 2
(d)noneofthese
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303
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
TwoparalleltangentstoagivencirclearecutbyathirdtangentatthepointsIf isthecenterofthegivencircle,then dependsontheradiusofthecircle.depends on the center of the circle. depends on the slopes of three tangents. isalwaysconstant
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304
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Three equal circles each of radius touch one another. The radius of the circle
touching all the three given circles internally is (b)
(d)
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305
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
If the radius of the circumcircle of the triangle TPQ, where PQ is chord of contactcorresponding topointTwith respect to circle , is 6units,thenminimumdistancesofTfromthedirectorcircleofthegivencircleis
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(g2 + f2 − c)4ch2
(g2 − c)2(g2 + f2 − c)
4ch2
(f2 − c)2(g2 + f2 − c)
4ch2
(f 2 − f2)2
AandB.C ∠ACB
r
(2 + √3)r r(2 + √3)
√3
r(2 − √3)
√3(2 − √3)r
x2 + y2 − 2x + 4y − 11 = 0
306
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
The equation of the locus of the middle point of a chord of the circle such that the pair of lines joining the origin to the point of
intersectionofthechordandthecircleareequallyinclinedtothex-axisis(b) (d)noneofthese
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307
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Theequationofcircleofminimumradiuswhichcontactsthethreecircle
thentheradiusofgivencircleis thenthevalueofl+mis:
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308
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
If the length of the common chord of two circles andis , thenthevaluesof are (b) (c) (d)
noneofthese
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309
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
A triangle is inscribed in a circle of radius1.Thedistancebetween theorthocentreandthecircumcentreofthetrianglecannotbe
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310
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Acircle ofradiusbtouchesthecircle externallyandhasitscentreon thepositiveX-axis;anothercircle of radiusc touches thecircle , externallyandhasitscentreonthepositivex-axis.Given thenthreecircleshaveacommontangentifa,b,carein
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x2 + y2 = 2(x + y)x + y = 2
x − y = 2 2x − y = 1
x2 + y2 − 4y − 5 = 0, x2 + y2
+ 12x + 4y + 31 = 0, x2 + y2
+ 6x + 12y + 36 = 0(l+ √949)m
36
x2 + y2 + 8x+ 1 = 0x2 + y2 + 2μy − 1 = 0 2√6 μ ±2 ±3 ±4
C1 x2 + y2 = a2
C2 C1a < b < c
311
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Let , , be circles of radii 5,3,2 respectively. and , touch each otherexternallyandCinternally.Acircle touches and externallyandCinternally.Ifitsradiusis wheremandnarerelativelyprimepositiveintegers,then2n-mis:
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312
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If the circle lies completely inside the circle then a) b) c)
d)
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313
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Considerthecircle .LetObethecentreofthecircleand tangent at A(7,3) and B(5, 1) meet at C. Let S=0 represents family of circlespassingthroughAandB,then
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314
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
Tangent drawn from the point to the circle will beperpendiculartoeachotherif equals5(b) (c)4(d)
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
C C1 C2 C1 C2C3 C1 C2
m
n
x2 + y2 + 2a1x + c = 0x2 + y2 + 2a2x+ c = 0 a1a2 > 0, c < 0 a1a2 > 0, c > 0a1a2 < 0, c < 0 a1a2 < 0, c > 0
x2 + y2 − 10x − 6y + 30 = 0
(a, 3) 2x2 + 2y2 = 25a −4 −5
1 1
315isapointonthecircle and isanotherpointon the
circle such that are length units. Then, the coordinates of can be
(b) (d)noneofthese
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316
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Let berealvariablessatisfying .Let
and . Then (b)
(d)
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317
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Threesidedofatrianglehaveequations Then where is the equation of the
circumcircleofthetriangleif
noneofthese
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
A( , )1
√2
1
√2x2 + y2 = 1 B
AB =π
2B
( , 1√2)1
√2( − , 1√2)1
√2( − , − )1
√2
1
√2
xandy x2 + y2 + 8x− 10y − 40 = 0a =
max {√(x + 2)2 + (y − 3)2}b = min {√(x + 2)2 + (y − 3)2} a + b = 18 a + b = √2
a − b = 4√2 a.b = 73
L1 ≡ y −mix = o; i = 1, 2and3.L1L2 + λL2L3 + μL3L1 = 0 λ ≠ 0, μ ≠ 0,
1 + λ + μ = m1m2 + λm2m3
+ λm3m1m1(1 + μ) +m2(1 + λ)
+ m3(μ + λ) = 0
+ + = 1 + λ + μ1m3
1m1
1m1
318
Iftheequation
representsacircle,thentheconditionforthatcircletopassthroughthreequadrantsonlybutnotpassingthroughtheoriginis (b) (d)
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319
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
The points on the line from which the tangents drawn to the circleareatrightanglesis(are)(a) (b) (c)
(d)
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320
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Co-ordinatesof thecentreofacircle,whose radius is2unitandwhich touches thepairoflinesines is(are)
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321
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If the circles and touch each
other,then is (b)0(c)1(d)
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322
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
PointMmovesonthecircle .Thenitbrokesawayfromitandmovingalonga tangent to thecircle,cuts thex-axisat thepoint (-2,0).Theco-ordinatesofapointonthecircleatwhichthemovingpointbrokeawayis
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323
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Theequationofthetangenttothecircle passingthrough is(b) (d)
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x2 + y2 + 2hxy + 2gx + 2fy + c= 0
f 2 > c g 2 > 2 c > 0 h = 0
x = 2x2 + y2 = 16 (2, 2√7) (2, 2√5) (2, − 2√7)
(2, − 2√5)
x2 − y2 − 2x + 1 = 0
x2 + y2 − 9 = 0 x2 + y2 + 2ax + 2y + 1 = 0
α −43
43
(x − 4)2 + (y − 8)2 = 20
x2 + y2 = 25 ( − 2, 11)4x+ 3y = 25 3x + 4y = 38 24x − 7y + 125 = 0 7x + 24y = 250
324
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If the area of the quadrilateral by the tangents from the origin to the circleandtheradiicorrespondingtothepointsofcontactis
thenavalueof is9(b)4(c)5(d)25
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325
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
The equation of the circle which touches the axes of coordinates and the line and whose center lies in the first quadrant is
,where is(a)1(b)2(c)3(d)6
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326
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Theequationsoftangentstothecircle drawnfromtheoriginin (b) (c) (d)
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327
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Ifacirclepasses through thepointof intersectionof the lines andwiththecoordinateaxis,thenvalueof is
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
26Which of the following lines have the intercepts of equal lengths on the circle, (A) (B) (C)
x2 + y2 + 6x − 10y + c = 015, c
+ = 1x
3y
4x2 + y2 − 2cx − 2cy + c2 = 0 c
x2 + y2 − 6x− 6y + 9 = 0x = 0 x = y y = 0 x + y = 0
x+ y + 1 = 0x + λy − 3 = 0 λ
x2 + y2 − 2x + 4y = 0 3x − y = 0 x + 3y = 0 x + 3y + 10 = 03 − − 10 = 0
328 (D)
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329
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
The equation of the line(s) parallel to which touch(es) the circle is (are) (a) (b)
(c) (d)3
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330
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Thecircles and touchinternally touch externally have as the common tangent at thepoint of contact have as the common tangent at the point ofcontact
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331
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
The circles and a)aresuchthatthenumberofcommontangentsonthemis2b)areorthogonalc)are
suchthatthelengthoftheircommontangentsis d)aresuchthatthelength
oftheircommonchordis
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
3x − y − 10 = 0
x − 2y = 1x2 + y2 − 4x − 2y − 15 = 0 x − 2y + 2 = 0 x − 2y − 10 = 0x − 2y − 5 = 0 x − y − 10 = 0
x2 + y2 − 2x − 4y + 1 = 0 x2 + y2 + 4x + 4y − 1 = 03x + 4y − 1 = 0
3x + 4y + 1 = 0
x2 + y2 + 2x + 4y − 20 = 0 x2 + y2 + 6x − 8y + 10 = 0
5( )125
14
5√32
332A particle from the point moves on the circle and aftercoveringaquarterofthecircleleavesittangentially.Theequationofalinealongwiththepointmovesafterleavingthecircleis
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333
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
The equation of a circle of radius 1 touching the circles is
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334
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If the circles and touch each
other,then is (b)0(c)1(d)
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335
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
The rangeofvaluesof such that theangle between thepairof tangentsdrawnfrom tothecircle satisfies`pi/2WatchFreeVideoSolutiononDoubtnut
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Iftheconicswhoseequationsare
P(√3, 1) x2 + y2 = 4
x2 + y2 − 2|x| = 0x2 + y2 + 2√2x + 1 = 0 x2 + y2 − 2√3y + 2 = 0x2 + y2 + 2√3y + 2 = 0 x2 + y2 − 2√2 + 1 = 0
x2 + y2 − 9 = 0 x2 + y2 + 2ax + 2y + 1 = 0
α −43
43
a θ(a, 0) x2 + y2 = 1
S1 : (sin2 θ)x2 + (2h tanθ)xy
+ (cos2 θ)y2 + 32x+ 16y + 19
= 0
336
intersectatfourconcyclicpoints,where thenthecorrectstatement(s)can
be (b) (d)noneofthese
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337
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thecenter(s)ofthecircle(s)passingthroughthepoints(0,0)and(1,0)andtouching
the circle is (are) (a) (b) (c) (d)
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338
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Eachquestionhasfourchoicesa,b,candd,outofwhichonlyoneiscorrect.Eachquestion contains STATEMENT 1 and STATEMENT 2. Both the statements areTRUE and STATEMENT 2 is the correct explanation of STATEMENT1. Both thestatements are TRUE but STATEMENT 2 is NOT the correct explanation ofSTATEMENT 1. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE.STATEMENT 1 is FALSE and STATEMENT 2 is TRUE. Statement 1:
, where is order of matrix Statement 2:
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339
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Statement1:Ifthechordsofcontactoftangentsfromthreepoints tothecircle areconcurrent,then willbecollinear.Statement2:Lines
alwasypassthroughafixedpointfor .
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340
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Statement1:Circles and donothaveanycommon tangent. Statement 2 : If two circles are concentric, then they do not havcommontangents.
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S1 : (sin2 θ)x2 − (2h ′ cot θ)xy
+ (sin2 θ)y2 + 16x+ 32y + 19
= 0θ[0, ],π
2h + h ′ = 0 h − h ′ = 0 θ =
π
4
x2 + y2 = 9 ( , )32
12
( , )12
32
( , 2 )12
12
( , − 2 )12
12
|adj(adj(adjA))| − |A|n−1 ^ 3 n A.|adjA| = |A|n.
A,BandCx2 + y2 = a2 A,BandC
(a1x + b1y + c1) + k(a2x + b2y
+ c2) = 0k ∈ R
x2 + y2 = 144 x2 + y2 − 6x − 8y = 0
341
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Statement1:Theleastandgreatestdistancesofthepoint fromthecircleare6unitsand15units,respectively.Statement2:
A point lies outside the circle ifwhere
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342
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Statement1 :Thenumberofcirclespassing through(1,2), (4,8)and(0,0) isone.Statement2:Everytrianglehasonecircumcircle
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343
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Let bethecirclewithcenter andradius1and bethecirclewithcenterandradius2.Statement1:Circles alwayshave
at least one common tangent for any value of Statement 2 : For the two circleswhere aretheirradiiforanyvalueof
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344
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Tangentsaredrawnfromtheorigintothecircle
Statement 1 : Angle between the tangents is Statement 2 : The given circle is
P(10, 7)x2 + y2 − 4x − 2y − 20 = 0
(x1, y1) S = x2 + y2 + 2gx + 2fy + c = 0S1 > 0,S1 = x12 + y12 + 2gx1 + 2fy1 +⋅
C1 O1(0, 0) C2
O2(t, t2 + 1), (t ∈ R), C1andC2
tO1O2 ≥ |r1 − r2|, r1andr2 t.
x2 + y2 − 2hx − 2hy + h2 = 0,(h ≥ 0)
π
2
touchingthecoordinateaxes.
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345
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Consider two circles andStatement1:Boththecirclesintersecteachotherattwodistinctpoints.Statement2:Thesumofradiiofthetwocirclesisgreaterthanthedistancebetweentheircenters.
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346
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
From the point , tangents are drawn to the circleStatement1:Theareaofquadrilateral beingtheorigin)is4.
Statement2:Theareaofsquareis where isthelengthofside.
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347
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Normal To A CircleAtAGivenPoint
Statement 1 : The center of the circle having and as itsnormals is (1, 2) Statement 2 : The normals to the circle always pass through itscenter
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x2 + y2 − 4x− 6y − 8 = 0 x2 + y2 − 2x − 3 = 0
P(√2, √6) PAandPB
x2 + y2 = 4 OAPB(Oa2, a
x + y = 3 x − y = 1
348
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Statement 1 : The circle having equation intersectsboththecoordinateaxes.Statement2:Thelengthsof interceptsmadebythe
circle having equation are and
,respectively.
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349
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Statement1:Thechordofcontactofthecircle w.r.t.thepoints(2,3),(3, 5), and (1, 1) are concurrent. Statement 2 : Points (1, 1), (2, 3), and (3, 5) arecollinear.
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350
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Statement1 :Theequation represents, fordifferentvaluesof asystemofcirclespassingthroughtwofixedpointslyingonthex-axis.Statement 2 : is a circle and is a straight line. Thenrepresentsthefamilyofcirclespassingthroughthepointsofintersectionofthecircleandthestraightline(where isanarbitraryparameter).
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351
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
StatementIThechordofcontactoftangentfromthreepointsA,BandCtothecircleareconcurrent,thenA,BandCwillbecollinear.StatementIIA,Band
Calwayslieonthenormaltothecircle .
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352
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Statement1:Points
areconcyclic.Statement2:Points forman isosceles trapeziumor
meetat Then
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353
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Common Chord OfTwoCircles
Statement 1 : The equations of the straight lines joining the origin to the points ofintersection of and is
. Statement 2 : is the common chord ofand
x2 + y2 − 2x + 6y + 5 = 0xandy
x2 + y2 + 2gx + 2fy + c = 0 2√g2 − c2√f2 − c
x2 + y2 = 1
x2 + y2 − 2x − 2ay − 8 = 0a,
S = 0 L = 0 S + λL = 0
λ
x2 + y2 = a2
x2 + y2 = a2
A(1, 0), B(2, 3), C(5, 3),
andD(6, 0)A,B,C, andD
ABandCD E . EA.EB = EC
.ED.
x2 + y2 − 4x − 2y = 4 x2 + y2 − 2x − 4y − 4 = 0x − y = 0 y + x = 0x2 + y2 − 4x − 2y = 4 x2 + y2 − 2x − 4y − 4 = 0
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354
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
Statement 1 : Two orthogonal circles intersect to generate a common chord whichsubtendscomplimentaryanglesattheircircumferences.Statement2:Twoorthogonalcircles intersect togenerateacommonchordwhichsubtendssupplementaryanglesattheircenters.
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355
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
Statement-1: The point does not lie outside the parabola
when Statement-2: The point
liesoutsidetheparabola if .
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356
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Statement 1 : If the circle with center cut the circlesand , then both the intersections are
orthogonal.Statement2:Thelengthoftangentfrom for isthesameforboththegivencircles.
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Statement 1 : Let and
(sin α, cos aα)
y2 + x− 2 = 0 α ∈ [ , (5 ] ∪ [π, ]π
2π
63π2
(x1, y1) y2 = 4ax y21 − 4ax1, 0
P(t, 4 − 2t), t ∈ R,x2 + y2 = 16 x2 + y2 − 2x − y − 12 = 0
P t ∈ R
S1 : x2 + y2 − 10x − 12y − 39 = 0,S2x
2 + y2 − 2x − 4y + 1 = 0
357
Theradicalcenterofthesecirclestakenpairwiseis Statement2 :Thepointofintersectionofthreeradicalaxesofthreecirclestakeninpairsisknownastheradicalcenter.
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358
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Statement1:Theequationofchordthroughthepoint whichisfarthestfromthe center of the circle is .Statement1:Innotations,theequationofsuchchordofthecircle bisectedat
mustbe
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359
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Statement 1 : If two circles and touch each other, then Statement 2 :
Twocirclestouchotherifthelinejoiningtheircentersisperpendiculartoallpossiblecommontangents.
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360
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Statement 1 :The circles and
touch if Statement2 :Twocenters and radii
respectively,toucheachotherif
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
S3 : 2x2 + 2y2 − 20x = 24y + 78= 0.
( − 2, − 3).
( − 2, 4)x2 + y2 − 6x + 10y − 9 = 0 x + y − 2 = 0
S = 0(x1, y1) T = S.
x2 + y2 + 2gx + 2fy = 0x2 + y2 + 2g ′ x+ 2f ′ y = 0 f ′ g = fg ′ .
x2 + y2 + 2px + r = 0 x2 + y2 + 2qy + r = 0
+ = .1p2
1q2
1e
C1andC2 r1andr2,
|r1 ± r2| = c1c2.
361Let and be two circles whose equations are and
and isavariablepoint
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362
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Let the lines and intersect at rightangles at (where are parameters). If the locus of is
,thenthevalueof is__________
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363
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Consider the family of circles passing through twofixedpoints .Thenthedistancebetweenthepoints is_____
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364
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
If realnumbers satisfy then theminimum
valueof is_________
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365
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Theline intersectsthecurve atpoints .Thecircle on as diameter passes through the origin. Then the value ofis__________
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C1 C2 x2 + y2 − 2x = 0x2 + y2 + 2x = 0 P(λ, λ)
(y − 2) = m1(x − 5) (y + 4) = m2(x − 3)P m1andm2 P
x2 + y2gx + fy + 7 = 0 |f + g|
x2 + y2 − 2x − 2λy − 8 = 0AandB AandB
xandy (x + 5)2 + (y − 12)2 = (14)2,
√x2 + y2
3x + 6y = k 2x2 + 3y2 = 1 AandBAB k2
366
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
The sum of the slopes of the lines tangent to both the circles andis________
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367
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Director Circle AndItsEquation
Acircle isthedirectorcircleofthecircleisthedirectorcircleofcircle andsoon.Ifthesumofradiiofallthesecirclesis2,thenthevalueof is ,wherethevalueof is___________
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368
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Two circle are externally tangent. Lines and are common tangentswith on the smaller circle and the on the larger circle. If
thenthesquareoftheradiusofthecircleis___________
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369
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If the lengthofacommoninternal tangent to twocircles is7,andthatofacommonexternaltangentis11,thentheproductoftheradiiofthetwocirclesis
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370
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Line segments and are diameters of the circle of radius one. If,thelengthoflinesegment is_________
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371
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Asshown in the figure, threecircleswhichhave thesame radius r,havecentresat. If theyhaveacommon tangentline,asshown then, their
radius'r'is-
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x2 + y2 = 1(x − 6)2 + y2 = 4
x2 + y2 + 4x − 2√2y + c = 0 S1andS1S2,
c k√2 k
PAB PA'B'AandA' B ′andB'
PA = AB = 4,
AC BD∠BDC = 600 AB
(0, 0); (1, 1) and (2, 1)
372
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
The acute angle between the line and the circleis .Then
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373
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If two perpendicular tangents can be drawn from the origin to the circle,thenthevalueof is___
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374
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Let Numberofpoints oncircle suchthatareaoftrianglewhosevertiesareA,B,Cispositiveintegeris:
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375
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Ifthecircle
and toucheachother,thenthemaximumvalueofcis
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376
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Twocircles bothpass through thepoints and touchtheline at respectively.Thepossiblecoordinatesofapoint
suchthatthequadrilateral isaparallelogram,are Thenthevalueof is_________
3x − 4y = 5x2 + y2 − 4x + 2y − 4 = 0 θ 9 cosθ =
x2 − 6x+ y2 − 2py + 17 = 0 |p|
A( − 4, 0),B(4, 0) c = (x, y) x2 + y2 = 16
x2 + y2 + (3 + sin β)x + 2cos αy= 0x2 + y2 + 2cos αx + 2cy = 0
C1andC2 A(1, 2)andE(2, 1)4x − 2y = 9 BandD,
C, ABCD (a, b).|ab|
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377
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Difference invaluesof theradiusofacirclewhosecenter isat theoriginandwhichtouchesthecircle is_____________
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378
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Find the equation of the circle whose radius is which touches the circleexternallyatthepoint
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379
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Findtheequationofacirclewhichpassesthroughthepoint andwhosecentreisthelimitofthepointofintersectionofethlines
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380
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
.LetAbe thecentreof thecircle Suppose that thetangentsat thepointsB(1,7)andD(4,-2)on thecirclemeetat thepointC.Find theareaofthequadrilateralABCD
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x2 + y2 − 6x − 8y + 21 = 0
5andx2 + y2 − 2x − 4y − 20 = 0 (5, 5).
(2, 0)
3x+ 5y = 1and(2 + c)x + 5c2y
= 1asc→1 .
x2 + y2 − 2x − 4y − 20 = 0
381
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Findtheequationsofthecirclespassingthroughthepoint andtouchingthelines and
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382
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Throughafixedpoint(h,k)secantsaredrawntothecircle .Thenthelocusofthemid-pointsofthesecantsbythecircleis
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383
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
If theabscissaandordinatesof twopoints are the roots of the equationsand , respectively, then find theequation
ofthecirclewith asdiameter.
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384
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Lines and touchacircleC1ofdiameter6.IfthecentreofC1,liesinthefirstquadrantthentheequationofthecircleC2,whichisconcentricwithC1,andcutsinterceptsoflength8ontheselinesis
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385
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Letagiven line intersect theXandYaxesatPandQrespectively.Letanotherline perpendicularto cuttheXandY-axesatRandS,respectively.ShowthatthelocusofthepointofintersectionofthelinePSandQRisacirclepassingthroughtheorigin
( − 4, 3)x + y = 2 x− y = 2
x2 + y2 = r2
PandQx2 + 2ax− b2 = 0 x2 + 2px − q2 = 0
PQ
5x + 12y − 10 = 0 5x − 12y − 40 = 0
L1L2 L1
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386
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thecircle isinscribedinatrianglewhichhastwoofitssides along the coordinate axes. The locus of the circumcenter of the triangle is
.Find
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387
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Let beagivencircle.FindthelocusofthefootoftheperpendiculardrawnfromtheoriginuponanychordofSwhichsubtendsarightangleattheorigin.
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388
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
If(m_i,1/m_i),i=1,2,3,4areconcyclicpointsthenthevalueof is
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389
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
A circle touches the line at point P such that , Circle contains(-10,2) in its interior& lengthof itschordon the line is .Determinetheequationofthecircle
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
x2 + y2 − 4x − 4y + 4 = 0
x + y − xy + k(x2 + y2) = 012 k.
S ≡ x2 + y2 + 2gx + 2fy + c =
m1m2m3m4
y = x OP = 4√2x + y = 0 6√2
390 Twocircles,eachofradius5units,toucheachotherat(1,2).Iftheequationoftheircommontangentsis ,findtheequationsofthecircles.
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391
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Letacirclebegivenby
.Find the condition on if two chords each bisected by the x-axis, can be
drawntothecirclefrom
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392
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Considera familyof circlespassing through thepoints (3, 7) and (6,5).Answer thefollowingquestions.Numberofcircleswhichbelongtothefamilyandalsotouchingx-axisare
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393
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Find thecoordinatesof thepointatwhich thecirclesand touch each other. Also, find equations ofcommontangentstouchingthecirclesthedistinctpoints.
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394
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Findtheintervalsofthevaluesof forwhichtheline bisectstwochords
drawnfromthepoint tothecircle
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395
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Consider a curve and a point P not on the curve. A linedrawnfromthepointPintersectthecurveatpointsQandR.Ifheproduct isindeopendentoftheslopeofthelinethenshowthatthecurveisacircle
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4x + 3y = 10
2x(x − a) + y(2y − b) = 0,
(a ≠ 0, b ≠ 0)a and b
(a, )b2
x2 − y2 − 4x − 2y + 4 = 0x2 + y2 − 12x − 8y + 36 = 0
a y + x = 0
( , )1 + √2a2
1 − √2a2
2x2 + 2y2 − (1 + √2a)x
− (1 − √2a) = 0
ax2 + 2hxy + by2 = 1PQ. PR
396
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Let beanycirclewithcentre Provethatatmosttworationalpointscanbethere on (A rational point is a point both of whose coordinates are rationalnumbers)
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397
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
C1andC2aretwoconcentriccircles,theradiusofC2beingtwicethatofC1.FromapointPonC2,tangentsPAandPBaredrawntoC1.ThenthecentroidofthetrianglePAB(a)liesonC1(b)liesoutsideC1(c) liesinsideC1(d)maylieinsideoroutsideC1butneveronC1
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398
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Let andbe two tangentsdrawn from(-2,0)onto thecircle .DeterminethecirclestouchingCandhaving astheirpairoftangents.Further,findtheequationsofallpossiblecommontangentstothesecircleswhentakentwoatatime
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399
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Let betheequationofpairoftangentsdrawnfromtheorigintoacircleofradius3,withcenter inthefirstquadrant. IfA isthepointofcontact.FindOA
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C (0, √2).C .
T1, T2 C :x2 + y2 = 1T1, T2
2x2 + y2 − 3xy = 0
400
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Tangents are drawn from external poinl P(6,8) to the, circle find theradius rof thecirclesuch thatareaof triangle formedby the tangentsandchordofcontactismaximumis
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401
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
The radiusof theof circle touching the line at (1,-1) and cuttingorthogonallythecirclehavinglinesegmentjoining(0,3)and(-2,-1)asdiameteris
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402
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Two circles intersect at two distinct points in a line passingthrough meetscircles at ,respectively.Let bethemidpointof
meetscircles at , respectively.Thenprovethatisthemidpointof
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403
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thetwopoints and inaplanearesuchthatforallpoints liesoncirclesatisfied
,then willnotbeequalto
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404
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
The points of intersection of the line and the circleare________and________
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x2 + y2 = r2
2x + 3y + 1 = 0
C1andC2 PandQP C1andC2 AandB Y
AB, andQY C1andC2 XandZ YXZ .
A B P
P B = kA
Pk
4x − 3y − 10 = 0x2 + y2 − 2x + 4y − 20 = 0
405
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
If the lines and are tangents toacircle, thenfindtheradiusofthecircle.
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406
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
findtheareaof thequadrilateral formedbyapairof tangents fromthepoint(4,5) tothecircle andpairofitsradii.
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407
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Fromtheorigin,chordsaredrawntothecircle .Theequationofthelocusofthemid-pointsofthesechordsiscirclewithradius
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408
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Theequationofthelinepassingthroughthepointsofintersectionofthecircles
is
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CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
3x − 4y + 4 = 0 6x − 8y − 7 = 0
x2 + y2 − 4x− 2y − 11 = 0
(x − 1)2 + y2 = 1
3x2 + 3y2 − 2x + 12y − 9
= 0 and x2 + y2 + 6x + 2y − 15= 0
409
FromthepointA(0,3)onthecircle achordABisdrawn&extended to a M point such that AM=2AB. The equation of the locus of M is: (A)
(B) (C) (D)
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410
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Findtheareaofthetriangleformedbythetangentsfromthepoint(4,3)tothecircleandthelinejoiningtheirpointsofcontact.
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411
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
If the circle intersects another circle of radius in such a
manner that,thecommonchord isofmaximum lengthandhasaslopeequal to ,
thentheco-ordinatesofthecentreof are:
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412
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Theareaofthetriangleformedbythepositivex-axiswiththenormalandthetangenttothecircle at is
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413
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
If a circle passes through the points of intersection of the coordinate axeswith thelines and ,thenthevalueof is............
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x2 + 4x+ (y − 3)2 = 0
x2 + 8x+ y2 = 0 x2 + 8x + (y − 3)2 = 0 (x − 3)2 + 8x + y2 = 0x2 + 8x+ 8y= 0
x2 + y2 = 9
C1 :x2 + y2 = 16 C2 534
C2
x2 + y2 = 4 (1, √3)
λx − y + 1 = 0 x − 2y + 3 = 0 λ
414
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
The equation of the locus of the mid-points of chords of the circlethatsubtendsanangleofatitscentreis atits
centreis thenkis
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415
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Intersection Of ALineAndACircle
Theinterceptontheline bythecircle isAB.EquationofthecirclewithABasadiameteris
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416
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Twoverticesofanequilateraltriangleare and(1,0),anditsthirdvertexliesabovethex-axis.Theequationofitscircumcircelis____________
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417
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
The chords of contact of the pair of tangents drawn from each point on the linetothecircle passthroughthepoint(a,b)then4(a+b)is
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418
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Position Of PointsWithRespectToACircle
No tangentcanbedrawn from thepoint to thecircumcircleof the triangle
withvertices .
4x2 + 4y2 − 12x+ 4y + 1 = 0 2π
3x2 + y2 − kx + y + = 0
3116
y = x x2 + y2 − 2x = 0
( − 1, 0)
2x+ y = 4 x2 + y2 = 1
( , 1)52
(1, √3), (1, − √3), (3, − √3)
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419
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Theline isadiameterofthecircle
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420
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
A square is inscribed in the circle . Its sides areparallel to the coordinate axes. One vertex of the square is (b)
(d)noneofthese
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421
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_FamilyOfCircle
Twocircle and aregiven.Then theequationof the circle through their points of intersection and the point (1, 1) is
none ofthese
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422
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
Thelocusofthemidpointofachordofthecircle whichsubtendsarightangleattheoriginsis (b) (d)
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
If a circle passes through the point and cuts the circle
x + 3y = 0 x2 + y2 − 6x + 2y = 0
x2 + y2 − 2x + 4y + 3 = 0(1 + √2, − 2)
(1 − √2, − 2) (1, − 2 + √2)
x2 + y2 = 6 x2 + y2 − 6x+ 8 = 0
x2 + y2 − 6x + 4 = 0 x2 + y2 − 3x + 1 = 0 x2 + y2 − 4y + 2 = 0
x2 + y2 = 4x + y = 2 x2 + y2 = 1 x2 + y2 = 2 x + y = 1
(a, b) x2 + y2 = k2
423 orthogonally,thentheequationofthelocusofitscenteris
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424
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
Ifthetwocircles
intersectintwodistinctpoint,then
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425
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Definition
Thelines and arethediametersofacircleofarea154sq. units. Then the equation of the circle is
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426
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
The centre of a circle passing through (0,0), (1,0) and touching the CircIe
isa. b. c. d.
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427
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
The locus of the centre of a circle which touches externally the circleandalsotouchesY-axis,isgivenbytheequation(a)
x2-6x-10y+14=0(b)x2-10x-6y+14=0(c)yr_6x-10y+14-0(d)y,2-10x-6y+14=0
(x − 1)2 + (y − 3)2 = r2 and x2
+ y2 − 8x + 2y + 8 = 0
2x − 3y = 5 3x − 4y = 7x2 + y2 + 2x − 2y = 62
x2 + y2 + 2x − 2y = 47 x2 + y2 − 2x + 2y = 47 x2 + y2 − 2x + 2y = 62
x2 + y2 = 9 ( , √2)12
( , )12
3
√2( , )32
1
√2( , − )12
1
√2
x2 + y2 − 6x − 6y + 14 = 0
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428
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Problems Based OnLocus
Theanglebetweenthepairoftangentsdrawnfromapoint tothecircle
is .thentheequationofthelocusofthepoint is
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429
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
If twodistinctchords,drawn from thepoint (p,q)on thecircle(where arebisectedbythex-axis,then (b) (d)
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430
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_RadicalAxis
An acute triangle PQR is inscribed in the circle . If Q and R havecoordinates(3,4)and(-4,3)respectively,thenfind .
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431
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_OrthogonalIntersectionOfCircles
Ifthecircles
intersectorthogonallythenkequals
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Px2 + y2 + 4x − 6y + 9 sin2 α
+ 13 cos2 α = 0
2α P x2 + y2 + 4x − 6y + 4 = 0x2 + y2 + 4x − 6y − 9 = 0 x2 + y2 + 4x − 6y − 4 = 0x2 + y2 + 4x − 6y + 9 = 0
x2 + y2 = px + qypq ≠ q) p2 = q2 p2 = 8q2 p2 < 8q2
p2 > 8q2
x2 + y2 = 25∠QPR
x2 + y2 + 2x + 2ky + 6
= 0 and x2 + y2 + 2ky + k = 0
432
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Let beachordofthecircle subtendingarightangleatthecenter.Then the locus of the centroid of the as moves on the circle is (1) Aparabola(2)Acircle(3)Anellipse(4)Apairofstraightlines
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433
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Let betangentattheextremitiesofthediameter ofacircleofradiusIf intersectatapoint onthecircumferenceofthecircle,thenprove
that .
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434
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
If the tangentat thepointon thecircle meets thestraightine atapointQonthey-axisthenthelengthofPQis
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435
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Thecentreofcircleinscribedinasquareformedbylines
is
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CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Different Cases OfTwoCircles
AB x2 + y2 = r2
ΔPAB P
PQandRS PRr. PSandRQ X
2r = √PQxRS
x2 + y2 + 6x + 6y = 25x − 2y + 6 = 0
x2 − 8x+ 12 = 0andy2 − 14y + 45= 0(4, 7) (7, 4) (9, 4) (4, 9)
2 2
436 Ifoneof thediametersof thecircle isachord to thecirclewithcentre(2,1),thentheradiusofcircleis:1. (2) (3)3(4)2
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437
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
A circle is given by , another circle C touches it externally andalsothex-axis,thenthelocusofcenteris:
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438
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Equation Of CircleWithTypicalConditions
Thecirclepassingthroughthepoint(-1,0)andtouchingthey-axisat(0,2)alsopassesthroughthepoint(A)(-3/2,0)(B)(-5/2,2)(C)(-3/2,5/2)(D)(-4,0)‘..22)
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439
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_ChordOfContact
Thelocusofthemid-pointofthechordofcontactoftangentsdrawnfrompointslyingonthestraightline tothecircle is:
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440
CENGAGE_MATHS_COORDINATE GEOMETRY_CIRCLES_Tangent To A CircleAtAGivenPoint
Tangentsaredrawnfromthepoint(17,7)tothecircle ,Statement IThe tangentsaremutuallyperpendicularStatement, llsThe locusof thepoints frornwhich mutually perpendicular tangents can be drawn to the given circle is
(a)Statement I is correct,Statement II is correct;Statement II isacorrect explanation forStatementl (b(Statement I is correct, Statement I| is correctStatement II is not a correct explanation for Statementl (c)Statement I is correct,StatementIIisincorrect(d)StatementIisincorrect,StatementIIiscorrect
x2 + y2 − 2x − 6y + 6 = 0√3 √2
x2 + (y − 1)2 = 1
4x− 5y = 20 x2 + y2 = 9
x2 + y2 = 169
x2 + y2 = 338
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441
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
Consider: where isa realnumberand Statement1 : If line isachordofcircle thenline isnotalwaysadiameterofcircle Statement2:Iflineisaadiameterofcircle thenline isnotachordofcircle Boththestatementare True and Statement 2 is the correct explanation of Statement 1. Both thestatement are True but Statement 2 is not the correct explanation of Statement 1.Statement1isTrueandStatement2isFalse.Statement1isFalseandStatement2isTrue.
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442
CENGAGE_MATHS_COORDINATEGEOMETRY_CIRCLES_Miscellaneous
The straight line 2x − 3y = 1 divides the circular region into twoparts.IfS={
},thenthenumberofpoint(s)inSlyinginsidethesmallerpartis
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L1 : 2x + 3y + p − 3 = 0 L2 : 2x + 3y + p + 3 = 0 pC : x2 + y2 + 6x − 10y + 30 = 0 L1
C, L2 C. L1C, L2 C.
x2 + y2≠¤6
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