co-ordinate geometry of the circle notes
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Co-ordinate Geometry of the Circle Notes. Aidan Roche 2009. Given the centre and radius of a circle, to find the equation of Circle K?. K. Method Sub centre & radius into: (x – h) 2 + (y – k) 2 = r 2 If required expand to: x 2 + y 2 +2gx +2fy + c = 0. r. c(h, k). - PowerPoint PPT PresentationTRANSCRIPT
(c) Aidan Roche 2009 1
Co-ordinate Geometry of the CircleNotes
Aidan Roche2009
(c) Aidan Roche 2009 2
Given the centre and radius of a circle, to find the equation of Circle K?
K
rMethod• Sub centre & radius into:
(x – h)2 + (y – k)2 = r2 • If required expand to:
x2 + y2 +2gx +2fy + c = 0c(h, k)
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To find the centre and radius. Given the Circle K: (x – h)2 + (y – k)2 = r2
Method• Centre: c(h, k)• Radius = r
Kr
c
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To find the centre and radius. Given the Circle K: x2 + y 2 = r2
Method• Centre: c(0, 0)• Radius = r
Kr
c
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To find centre and radius of K. Given the circle K: x2 + y2 +2gx +2fy + c = 0?
KMethod• Centre: c(-g, -f)
• Radius:
r
ccfgr 22
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Given equation of circle K, asked if a given point is on, inside or outside the circle?
a Method• Sub each point into the
circle formula K = 0
Answer > 0 outsideAnswer = 0 onAnswer < 0inside
b
c
K
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Important to remember
Theorem • Angle at centre is
twice the angle on the circle standing the same arc
cθ
2θa b
d
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Important to remember
Theorem • Angle on circle
standing the diameter is 90odiameter
90o
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To find equation of circle K given end points of diameter?
K Method• Centre is midpoint [ab]• Radius is ½|ab|• Sub into circle formula
a bcr
10
To prove a locus is a circle?
Method• If the locus of a set of
points is a circle it can be written in the form:
x2 + y2 +2gx + 2fy + c = 0• We then can write its
centre and radius.
c
K
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r
11
To find the Cartesian equation of a circle given Trigonometric Parametric equations?
Method• Trigonometric equations
of a circle are always in the form:x = h ± rcosѲy = k ± rsinѲ
• Sub h, k and r into Cartesian equation:(x – h)2 + (y – k)2 = r2
c
K
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r
12
To prove that given Trigonometric Parametric equations (x = h ± rcosѲ, y = k ± rsinѲ) represent a circle?
Method• Rewrite cosѲ (in terms of x, h & r)
and then evaluate cos2Ѳ.• Rewrite sinѲ (in terms of y, h & r)
and then evaluate sin2Ѳ.• Sub into: sin2Ѳ + cos2Ѳ = 1 • If it’s a circle this can be written
in the form: x2 + y2 +2gx + 2fy + c = 0
c
K
(c) Aidan Roche 2009
r
13
To find the Cartesian equation of circle (in the form: x2 + y2 = k) given algebraic parametric equations?
Method• Evaluate: x2 + y2
• The answer = r2
• Centre = (0,0) & radius = rc
K
(c) Aidan Roche 2009
r
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Given equations of Circle K and Circle C, to show that they touch internally?
K
Method• Find distance
between centres• If d = r1 - r2 QED
C
r1
r2
dc1
c2
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Given equations of Circle K and Circle C, to show that they touch externally?
K
Method• Find distance d
between centres• If d = r1 + r2 QED
C
r1
r2
d
c1
c2
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Given circle K and the line L to find points of intersection?
aMethod• Solve simultaneous equations
bL
K
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Important to remember
Theorem • A line from the
centre (c) to the point of tangency (t) is perpendicular to the tangent.
c90o
Tangent
K
radiust
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Important to remember
Theorem • A line from the
centre perpendicular to a chord bisects the chord.
c90o
a
bradius
d
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Given equation of Circle K and equation of Tangent T, find the point of intersection?
KT
Method• Solve the simultaneous
equations
t
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Given equation of Circle K and asked to find equation of tangent T at given point t?
K
tMethod 1• Find slope [ct]• Find perpendicular slope of T• Solve equation of the line
c
T
Method 2• Use formula in log tables
21
To find equation of circle K, given that x-axis is tangent to K, and centre c(-f, -g) ?
X-axis
Method• On x-axis, y = 0 so t is (-f, 0)• r = |f|• Sub into circle formula
c(-g, -f)K
(c) Aidan Roche 2009
t(-g, 0)
r = |f|
22
To find equation of circle K, given that y-axis is tangent to K, and centre c(-f, -g) ?
y-axis
Method• On y-axis, x = 0 so t is (0, -g)• r = |g|• Sub into circle formula
c(-g, -f)
K
(c) Aidan Roche 2009
t(0, -f)
r = |g|
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Given equation of Circle K and equation of line L, how do you prove that L is a tangent?
KL
Method 2• Find distance from c to L
• If d = r it is a tangent
22
)()(ba
cfbgad
r
Method 1• Solve simultaneous
equations and find that there is only one solution
c
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Given equation of Circle K & Line L: ax + by + c = 0 to find equation of tangents parallel to L?
K
r
Method 1• Find centre c and radius r• Let parallel tangents be:
ax + by + k = 0• Sub into distance from point
to line formula and solve:c
L
T1
T2
22
)()(ba
kfbgar
r
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Given equation of Circle K and point p, to find distance d from a to point of tangency?
K
c
t
Method• Find r• Find |cp|• Use Pythagoras to find d
p
T
r
|cp|
d?
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Given equation of Circle K and point p, to find equations of tangents from p(x1,y1)?
K cp
T1
r
T2
r
Method 1• Find centre c and radius r• Sub p into line formula and write
in form T=0 giving: mx – y + (mx1 – y1) = 0
• Use distance from point to line formula and solve for m:
2211
1)()(1)(
m
ymxgfmr
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Given equation of Circle K and Circle C, to find the common Tangent T?
K
T
Method• Equation of T is:
K – C = 0
C
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Given equation of Circle K and Circle C, to find the common chord L?
K
L
C
Method• Equation of T is:
K – C = 0
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Given three points and asked to find the equation of the circle containing them?
aMethod• Sub each point into formula:
x2 + y2 + 2gx + 2fy + c = 0• Solve the 3 equations to find:
g, f and c, • Sub into circle formula
b
c
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Given 2 points on circle and the line L containing the centre, to find the equation of the circle?
a Method• Sub each point into the circle:
x2 + y2 + 2gx + 2fy + c = 0• Sub (-g, -f) into equation of L• Solve 3 equations to find: g, f and c, • Sub solutions into circle equation
b
L
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Given the equation of a tangent, the point of tangency and one other point on the circle, to find the equation of the circle?
a Method• Sub each point into the circle:
x2 + y2 + 2gx + 2fy + c = 0• Use the tangent & tangent point to
find the line L containing the centre.• Sub (-g, -f) into equation of L• Solve 3 equations to find: g, f and c, • Sub solutions into circle equation b T
L