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Co-ordinate Geometry of the Circle Notes - PowerPoint PPT Presentation

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 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)

• 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)

• 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)

• 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

• 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]

• 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

• 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

• Tangent

(c) Aidan Roche 2009

Important to remember intersection?

• Theorem

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

• b

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

• C

• T

• Method

• Equation of T is: K – C = 0

K

(c) Aidan Roche 2009

• 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