drill exercise abj sir straight lines - abhijit kumar jha · pdf filedrill exercise abj sir...

Drill exercise Abj sir Straight Lines

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Drill Exercise - 1

1. Find the distance between the pair of points, (a sin , �b cos ) and (�a cos , b sin ).

2. Prove that the points (2a, 4a) (2a, 6a) and (2a + 3 a, 5a) are the vertices of an equilateral triangle.

3. Which point on y-axis is equidistant from (2, 3) and (�4, 1) ?

Drill Exercise - 2

1. Find the ratio in which the line segment joining (2, �3) and (5, 6) is divided by (i) x-axis (ii) y-axis.

2. If three vertices of a parallelogram are (a + b, a � b), (2a + b, 2a�b), (a � b, a + b), then find the fourthvertex.

3. Find the coordinates of points on the line joining the points P(3, �4) and Q(�2, 5) that is twice as farfrom P as from Q.

Drill Exercise - 3

1. Find the orthocentre of the triangle formed by the lines xy = 0 and x + y = 1.

2. In a triangle ABC with vertices A(1, 2), B(2, 3) and C(3, 1) and A = cos�1

5

4,

B = C = cos�1

10

1 then find the circumcentre of the triangle ABC.

3. If G be the centroid and I be the incentre of the triangle with vertices A(�36, 7), B(20, 7) and

C(0, �8) and GI = 3

25205 then find the value of .

Drill Exercise - 4

1. The vertices of ABC are (�2, 1), (5, 4) and (2, �3) respectively. Find the area of the triangle and thelength of the altitude through A.

2. Prove that the points (a, 0), (0, b) and (1, 1) are collinear if a

1 +

b

1 = 1.

3. The four vertices of a quadrilateral are (1, 2), (�5, 6), (7, �4) and (k, �2) taken in order. If the areaof the quadrilateral is zero, find the value of k.

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Drill Exercise - 5

1. Find the locus of a point equidistant from the point (2, 4) and the y-axis.

2. Find the locus of a point, so that the join of (�5, 1) and (3, 2) subtends a right angle at the movingpoint.

3. If O is the origin and Q is a variable point on y2 = x. Find the locus of the mid-point of OQ.

Drill Exercise - 6

1. (a) Determine �x� so that the line passing through (3, 4) and (x, 5) makes 135º angle with thepositive direction of x-axis.

(b) If the line passing through the points (2, �5), (�5, �5) then prove that line will be parallel tox-axis and if the line passing through the points (6, 3), (6, �3) then prove that line will beperpendicular to the x-axis.

2. If is the angle of inclination of the line joining the points (7, �2) and (3, 1), then find the value ofsin and cos .

3. Using the method of slope, show that the following points are collinear(i) A(4, 8), B(5, 12), C(9, 28) (ii) A(16, �18), B (3, �6), C(�10, 6)

Drill Exercise - 7

1. Are the points (3, �4) and (2, 6) on the same or opposite sides of the line 3x � 4y = 8 ?

2. Which one of the points (1, 1), (�1, 2) and (2, 3) lies on the side of the line 4x + 3y � 5 = 0 on whichthe origin lies.

3. If the points (4, 7) and (cos , sin ), where 0 < < , lie on the same side of the line x + y � 1 = 0, thenprove that lies in the first quadrant.

Drill Exercise - 9

1. A straight line is drawn through the point P(2, 3) and is inclined at an angle of 30º with the x-axis. Findthe coordinates of two points on it at a distance 4 from P on either side of P.

2. Find the distance of the point (2, 3) from the line 2x � 3y + 9 = 0 measured along a line x � y + 1 = 0.

3. Find the equation of the line which passes through P(1, �7) and meets the axes at A and B respectivelyso that 4AP � 3BP = 0, where O is the origin.

Drill Exercise - 10

1. Show that if any line through the variable point A(k + 1, 2k), meets the lines 7x + y � 16 = 0,5x � y � 8 = 0, x � 5y + 8 = 0 at B, C, D respectively AC, AB and AD are in harmonic progression.(The three lines lie on the same side point A)




2. A straight line through the point (�2, �3) cuts the line x + 3y = 9 and x + y + 1 = 0 at B and Crespectively. Find the equation of the line if AB.AC = 20.

3. A line which makes an acute angle with the positive direction of x-axis is drawn through the pointP(3, 4) to cut the curve y2 = 4x at Q and R. Show that the lenghts of the segments PQ and PR arenumerical values of the roots of the equation r2 sin2 + 4r(2 sin � cos) + 4 = 0.

4. A straight lien through A(�15, �10) meets the liens x � y � 1 = 0, x + 2y = 5 and x + 3y = 7

respectively at A, B and C. If AB

12 +

AC

40 =

AD

52, prove that the line passed through the origin.

5. The base AB of a triangle ABC passes through the point (1, 5) which divides in it the ratio 2 : 1.If the equations of the sides AC and BC are 5x � y � 4 = 0 and 3x � 4y � 4 = 0 respectively, then findthe coordinates of the vertex A.

Drill Exercise - 11

1. Find the distance between the line 12 x � 5y + 9 = 0 and the point (2, 1).

2. If p is the length of the perpendicular from the origin to the line a

x +

b

y = 1, then prove that

2p

1 = 2a

1 + 2b

1.

3. If p and p be the perpendicular from the origin upon the straight lines x sec + y cosec = a and xcos � y sin = a cos 2. Prove that 4p2 + p2 = a2.

Drill Exercise - 12

1. Find the perpendicular distance of the point (1, 0) from the line 3x + 2y � 1 = 0, Also find theco-ordinate of the foot of perpendicular.

2. Find the image of the point (4, �13) in the line 5x + y + 6 = 0.

3. The point P() undergoes a reflection in the x-axis followed by a reflection in the y-axis. Show thattheir combined effect is the same as the single reflection of P() in the origin when > 0.

4. The image of the point A (1, 2) by the line mirror y = x is the point B and the image of B by the linemirror y = 0 is the point (). Find and .

5. The point P (4, 1) undergoes the following three transformations successively(i) reflection about the line y = x.(ii) translation through a distance 2 units along the positive direction of x-axis.(iii) rotation through an angle /4 about the origin in the anticlockwise direction.

Then find the coordinates of the final position.




Drill Exercise - 13

1. Find the equation of the straight line that passes through the point (3, 4) and perpendicular to the line3x + 2y + 5 = 0.

2. Find the equation of a straight line parallel to 2x + 3y + 11 = 0 and which is such that the sum of itsintercepts on the axes is 15.

3. Find the equation of the straight line which passes through the point (2, �3) and the point of intersectionof the lines x + y + 4 = 0 and 3x � y � 8 = 0.

Drill Exercise - 14

1. Prove that the lines 3x + y � 14 = 0, x � 2y = 0 and 3x � 8y + 4 = 0 are concurrent.

2. Find the value of , if the lines 3x � 4y � 13 = 0, 8x � 11 y � 33 = 0 and 2x � 3y + = 0 areconcurrent.

3. If the lines a1x + b

1y + 1 = 0, a

2x + b

2y + 1 = 0 and a

3x + b

3y + 1 = 0 are concurrent, show that the

points (a1, b

1), (a

2, b

2) and (a

3, b

3) are collinear.

Drill Exercise - 15

1. Find the angles between the pairs of straight lines

(i) x � y 3 � 5 = 0 and 3 x + y � 7 = 0

(ii) y = (2 � 3 ) x + 5 and y = (2 + 3 ) x � 7

2. Find the tangent of the angle between the lines which have intercepts 3, 4 and 1, 8 on the axesrespectively.

3. Prove that the straight lines (a + b) x + (a � b) y = 2ab, (a � b) x + (a + b) y = 2 ab and x + y = 0 form

an isosceles triangle whose vertical angle is 2 tan�1

b

a.

Drill Exercise - 16

1. Find the equation of the bisector of the angles between the straight lines 3x - 4y + 7 = 0 and12x - 5y - 8 = 0.

2. Find the equation of the obtuse angle bisector of lines 12x - 5y + 7 = 0 and 3y - 4x - 1 = 0.




3. Find the bisector of the acute angle between the lines 3x + 4y - 11 = 0 and 12 x - 5y - 2 = 0.Drill Exercise - 17

1. For the straight lines 4x + 3y � 6 = 0 and 5x + 12 y + 9 = 0 find the equation of the bisector of theangle which contains the origin.

2. Find the coordinates of the incentre of the triangle whose sides are x + 1 = 0, 3x � 4y � 5 = 0,5x + 12y � 27 = 0.

3. The sides of a triangle are x � y + 3 = 0, 7x � y + 3 = 0 and x + y + 1 = 0. Find the equations of theexternal bisectors of the angles at B and C. Also find the coordinates of the centre of the circleescribed to side BC.

Drill Exercise - 18

1. What will be the new coordinates of the point A (1, 2) if origin is shifted to the point at (�2, 3).

2. At what point the origin be shifted, if the coordinates of a point (4, 5) become (�3, 9) ?

3. If the axes are shifted to the point (1, �2) without rotation, what do the following equations become?(i) 2x2 + y2 � 4x + 4y = 0 (ii) y2 � 4x + 4y + 8 = 0

4. Shift the origin to a suitable point so that the equation y2 + 4y + 8x � 2 = 0 will not contain term in yand the constant term.

5. Verify that the area of the triangle with vertices (2, 3), (5, 7) and (�3, �1) remains invariant under thetranslation of the axes when the origin is shifted to the point (�1, 3)

Drill Exercise - 19

1. What will be the new coordinates of point A (1, 3) if coordinate axes is rotated by 45º in anticlockwisedirection.

2. What was the old coordinates of point A (2, 5) if coordinate axes is rotated by 30º in clockwisedirection.

3. If the axes be turned through an angle tan�1 2, what does the equation 4xy � 3x2 = a2 become ?

4. If (x, y) and (X, Y) be the coordinates of the same point referred to two sets of rectangular axes withthe same origin and if ax + by become pX + qY, where a, b are independent of x, y, prove thata2 + b2 = p2 + q2.

5. If the axes are shifted to the point (�2, �3) and then they are rotated through an angle of 45º inanticlockwise sense, what does the equation 2x2 + 4xy � 5y2 + 20x � 22y � 24 = 0 become ?

Drill exercise �20

1. For what value of does the equation 6x2 � 42xy + 60y2 � 11x + 10y + = 0 represent two straightlines ?

2. Prove that the angle between the straight lines given by,




(x cos � y sin )2 = (x2 + y2) sin2 is 2.

3. Show that the difference of the tangents of the angles which the lines,x2 (sec2 � sin2) � 2xy tan + y2 sin2 = 0 make with x-axis is 2.

4. Find the equation of the lines bisecting the angles between the pair of lines 3x2 + xy � 2y2 = 0.5. If the pairs of straight lines x2 � 2pxy � y2 = 0 and x2 � 2qxy � y2 = 0 be such that each pair bisects the

angle between the other pair, prove that pq = �1.

drill exercise �21

1. Find the equation of the lines joining the origin to the points of intersection of the line x + y = 1 withthe curve 4x2 + 4y2 + 4x - 2y - 5 = 0, and show that they are at right angles.

2. Find the condition that the pair of straight lines joining the origin to the intersection of the line,

y = mx + c and the curve x2 + y2 = a2 may be at right angles.

3. Find the equations of the lines joining the origin to the points of intersection of the curve

2x2 + 3xy - 4x + 1 = 0 and the line 3x + y = 1.

4. Show that the lines joining the origin to the points of intersection of the line gyfx and the curve

x2 + hxy - y2 + gx + fy + c = 0 are at right angles for all R if c = 0.

5. The line 0nmyx cuts the parabola y2 = 4ax at P and Q. Find the condition for OQOP where O is the origin.


1. Show that the straight line joining origin to the points of intersection of the curve

ax2 + 2hxy + by2 + 2gx = 0 and 0xg2ybxyh2xa 22 will be at right angles if

)ba(g)ba(g .

2. Prove that the angle between the lines joining the origin to the points of intersection of the line

y = 3x + 2 with the curve x2 + 2xy + 3y2 + 4x + 8y = 11 is

3

22tan 1 .

3. The circle x2 + y2 = a2 cuts off an intercept on the straight line 1myx , which subtends an angle

of 4

at the origin. Show that 2222222 2ma1ma4 .

4. A pair of straight lines drawn through the origin form an isosceles triangle right angled at the originwith the line 2x + 3y = 6. Find the equation of the pair of straight lines and the area of the triangle.

5. If the lines ax2 + 2h xy + by2 = 0 form two adjacent sides of a parallelogram and the line 1myn

is one diagonal, prove that the equation of the other diagonal is, h)x(amhm)y(b .




ANSWER - KEYDrill exercise �1

1. )ba(2 22

4

cos 3. (0, �1)

Drill exercise �21. (i) 1 : 2 externally (ii) 2 : 5 externally 2. (�b, b)

3. (�7, 14)Drill exercise �3

1. (0, 0) 2.

2,6

113.

25

1

Drill exercise �4

1. area = 20 sq. unit, length of altitude = 58

403. k = 3

Drill exercise �5

1. y2 � 8y � 4x + 20 = 0 2. x2 + y2 + 2x � 3y � 13 = 0

3. 2y2 = x

Drill exercise �6

1. (a) x = 2 2. sin = ± 5

3, cos = ±

5

4

Drill exercise �7

1. opposite sides 2. (�1, 2)Drill exercise �8

1. 3 x + y = 14 2. 5x + 2y + 6 = 0

3. x � 5y + 23 = 0, 7x + 4y � 8 = 0, 8x � y + 15 = 0

Drill exercise �9

1. (2 ± 2 3 , 3 ± 2) 2. 4 2




3. 28x � 3y = 49


2. x - y = 1, 3x - y + 3 = 0 5.

17

307,

17

75


1.13

28


1.13

2,

13

4,

13

72. (�1, �14)

4. 2, �1 5. (i) reflection of P is (1, 4) (iii)

2

7,

2

1


1. 2x � 3y + 6 = 0 2. 2x + 3y � 18 = 0 3. 2x � y � 7 = 0


2. = �7Drill exercise �15

1. (i) 90º (ii) 60º 2.7

4


1. 21 x + 27 y - 131 = 0, 99 x - 77 y + 51 = 0 2. 4x + 7y + 11 = 0

3. 11x + 3y - 17 = 0


1. 7x + 9y � 3 = 0 2.

3

2,

3

1




3. x + 2y � 6 = 0, 2x � y + 3= 0, (2, 1)Drill exercise �18

1 (3, �1) 2. (7, �4)

3. (i) 2X2 + Y2 = 6 (ii) Y2 = 4X 4.

2,

4

3


1. 2,22 2.

1

2

35,

2

53

3. X2 � 4Y2 = a2 5. X2 � 14XY � 7Y2 �12 = 0


1. = �10 4. x2 � 10xy � y2 = 0


1. 3x2 - 3y2 - 8xy = 0 2. 2c2 = a2(1 + m2)

3. x2 - y2 - 5xy = 0 5. 0na4


4. 5y2 + 24xy � 5x2 = 0, 2.77 sq. units.


drill exercise abj sir straight lines - abhijit kumar jha · pdf filedrill exercise abj sir...

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