Co-ordinate Geometry of the Circle Notes

(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)

(c) Aidan Roche 2009 3

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

(c) Aidan Roche 2009 4

To find the centre and radius. Given the Circle K: x2 + y 2 = r2

Method• Centre: c(0, 0)• Radius = r

Kr

c

(c) Aidan Roche 2009 5

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

(c) Aidan Roche 2009 6

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

(c) Aidan Roche 2009 7

Important to remember

Theorem • Angle at centre is

twice the angle on the circle standing the same arc

cθ

2θa b

d

(c) Aidan Roche 2009 8

Important to remember

Theorem • Angle on circle

standing the diameter is 90odiameter

90o

(c) Aidan Roche 2009 9

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

(c) Aidan Roche 2009

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

(c) Aidan Roche 2009

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

(c) Aidan Roche 2009 14

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

(c) Aidan Roche 2009 15

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

(c) Aidan Roche 2009 16

Given circle K and the line L to find points of intersection?

aMethod• Solve simultaneous equations

bL

K

(c) Aidan Roche 2009 17

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

(c) Aidan Roche 2009 18

Important to remember

Theorem • A line from the

centre perpendicular to a chord bisects the chord.

c90o

a

bradius

d

(c) Aidan Roche 2009 19

Given equation of Circle K and equation of Tangent T, find the point of intersection?

KT

Method• Solve the simultaneous

equations

t

(c) Aidan Roche 2009 20

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|

(c) Aidan Roche 2009 23

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

(c) Aidan Roche 2009 24

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

(c) Aidan Roche 2009 25

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?

(c) Aidan Roche 2009 26

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

(c) Aidan Roche 2009 27

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

(c) Aidan Roche 2009 28

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

(c) Aidan Roche 2009 29

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

(c) Aidan Roche 2009 30

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

(c) Aidan Roche 2009 31

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