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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).

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co ordinate geometry of the circle notes

Co-ordinate Geometry of the CircleNotes

Aidan Roche

2009

(c) Aidan Roche 2009

slide2

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 = r2
  • If required expand to: x2 + y2 +2gx +2fy + c = 0
  • r
  • c(h, k)

(c) Aidan Roche 2009

slide3

To find the centre and radius. Given the Circle K:(x – h)2 + (y – k)2 = r2

K

  • r
  • Method
  • Centre: c(h, k)
  • Radius = r
  • c

(c) Aidan Roche 2009

slide4

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

K

  • r
  • Method
  • Centre: c(0, 0)
  • Radius = r
  • c

(c) Aidan Roche 2009

slide5

To find centre and radius of K. Given the circle K: x2 + y2 +2gx +2fy + c = 0?

K

  • r
  • Method
  • Centre: c(-g, -f)
  • Radius:
  • c

(c) Aidan Roche 2009

slide6

Given equation of circle K, asked if a given point is on, inside or outside the circle?

  • a
  • b
  • Method
  • Sub each point into the circle formula K = 0
  • Answer > 0 outside
  • Answer = 0 on
  • Answer < 0 inside
  • c
  • K

(c) Aidan Roche 2009

slide7

Important to remember

  • d
  • Theorem
  • Angle at centre is twice the angle on the circle standing the same arc

θ

  • c

  • b
  • a

(c) Aidan Roche 2009

slide8

Important to remember

  • Theorem
  • Angle on circle standing the diameter is 90o

90o

  • diameter

(c) Aidan Roche 2009

slide9

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
  • r
  • b
  • a
  • c

(c) Aidan Roche 2009

slide10

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.
  • r
  • c
  • K

(c) Aidan Roche 2009

slide11

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 ± rcosѲ
  • y = k ± rsinѲ
  • Sub h, k and r into Cartesian equation:
  • (x – h)2 + (y – k)2 = r2
  • r
  • c
  • K

(c) Aidan Roche 2009

slide12

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
  • r
  • c
  • K

(c) Aidan Roche 2009

slide13

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 = r
  • r
  • c
  • K

(c) Aidan Roche 2009

slide14

Given equations of Circle K and Circle C, to show that they touch internally?

  • C
  • Method
  • Find distance between centres
  • If d = r1 - r2QED
  • r2
  • r1

K

  • c2
  • d
  • c1

(c) Aidan Roche 2009

slide15

Given equations of Circle K and Circle C, to show that they touch externally?

  • C
  • c2
  • Method
  • Find distance d between centres
  • If d = r1 + r2QED
  • r2

K

  • r1
  • d
  • c1

(c) Aidan Roche 2009

slide16

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

L

  • b
  • Method
  • Solve simultaneous equations
  • a
  • K

(c) Aidan Roche 2009

slide17

Important to remember

  • K
  • Theorem
  • A line from the centre (c) to the point of tangency (t) is perpendicular to the tangent.

90o

  • c
  • t
  • radius
  • Tangent

(c) Aidan Roche 2009

slide18

Important to remember

  • Theorem
  • A line from the centre perpendicular to a chord bisects the chord.
  • b
  • radius

90o

  • c
  • d
  • a

(c) Aidan Roche 2009

slide19

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

K

  • T
  • t
  • Method
  • Solve the simultaneous equations

(c) Aidan Roche 2009

slide20

Given equation of Circle K and asked to find equation of tangent T at given point t?

  • T
  • Method 1
  • Find slope [ct]
  • Find perpendicular slope of T
  • Solve equation of the line
  • t
  • c
  • Method 2
  • Use formula in log tables

K

(c) Aidan Roche 2009

slide21

To find equation of circle K, given that x-axis is tangent to K, and centre c(-f, -g) ?

  • Method
  • On x-axis, y = 0 so t is (-f, 0)
  • r = |f|
  • Sub into circle formula
  • K
  • c(-g, -f)
  • r = |f|
  • X-axis
  • t(-g, 0)

(c) Aidan Roche 2009

slide22

To find equation of circle K, given that y-axis is tangent to K, and centre c(-f, -g) ?

  • Method
  • On y-axis, x = 0 so t is (0, -g)
  • r = |g|
  • Sub into circle formula
  • r = |g|
  • t(0, -f)
  • c(-g, -f)
  • K
  • y-axis

(c) Aidan Roche 2009

slide23

Given equation of Circle K and equation of line L, how do you prove that L is a tangent?

  • Method 1
  • Solve simultaneous equations and find that there is only one solution
  • L

K

  • r
  • Method 2
  • Find distance from c to L
  • If d = r it is a tangent
  • c

(c) Aidan Roche 2009

slide24

Given equation of Circle K & Line L: ax + by + c = 0 to find equation of tangents parallel to L?

  • L
  • 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:
  • T1
  • r
  • c

K

  • r
  • T2

(c) Aidan Roche 2009

slide25

Given equation of Circle K and point p, to find distance d from a to point of tangency?

  • Method
  • Find r
  • Find |cp|
  • Use Pythagoras to find d
  • T
  • t
  • d?
  • r
  • p
  • c
  • |cp|

K

(c) Aidan Roche 2009

slide26

Given equation of Circle K and point p, to find equations of tangents from p(x1,y1)?

  • T1
  • 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:
  • p
  • r

K

  • c
  • r
  • T2

(c) Aidan Roche 2009

slide27

Given equation of Circle K and Circle C, to find the common Tangent T?

  • C
  • T
  • Method
  • Equation of T is: K – C = 0

K

(c) Aidan Roche 2009

slide28

Given equation of Circle K and Circle C, to find the common chord L?

  • L
  • Method
  • Equation of T is: K – C = 0
  • C

K

(c) Aidan Roche 2009

slide29

Given three points and asked to find the equation of the circle containing them?

  • b
  • Method
  • 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
  • a
  • c

(c) Aidan Roche 2009

slide30

Given 2 points on circle and the line L containing the centre, to find the equation of the circle?

  • b
  • 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

L

(c) Aidan Roche 2009

slide31

Given the equation of a tangent, the point of tangency and one other point on the circle, to find the equation of the circle?

  • L
  • 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

(c) Aidan Roche 2009

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