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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


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


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

K

  • r

  • Method

  • Centre: c(h, k)

  • Radius = r

  • c

(c) Aidan Roche 2009


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

K

  • r

  • Method

  • Centre: c(0, 0)

  • Radius = r

  • c

(c) Aidan Roche 2009


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

K

  • r

  • Method

  • Centre: c(-g, -f)

  • Radius:

  • c

(c) Aidan Roche 2009


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


Important to remember inside or outside the circle?

  • d

  • Theorem

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

θ

  • c

  • b

  • a

(c) Aidan Roche 2009


Important to remember inside or outside the circle?

  • Theorem

  • Angle on circle standing the diameter is 90o

90o

  • diameter

(c) Aidan Roche 2009


To find equation of circle K given end points of diameter? inside or outside the circle?

K

  • Method

  • Centre is midpoint [ab]

  • Radius is ½|ab|

  • Sub into circle formula

  • r

  • b

  • a

  • c

(c) Aidan Roche 2009


To prove a locus is a circle? inside or outside the 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


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


To prove that given Trigonometric Parametric equations 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


To find the Cartesian equation of circle Trigonometric Parametric equations?

(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


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


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


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

L

  • b

  • Method

  • Solve simultaneous equations

  • a

  • K

(c) Aidan Roche 2009


Important to remember intersection?

  • 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


Important to remember intersection?

  • Theorem

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

  • b

  • radius

90o

  • c

  • d

  • a

(c) Aidan Roche 2009


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


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


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


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


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


Given equation of Circle K & Line L: you prove that L is a tangent?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


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


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


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


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


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


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


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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