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Co-ordinate Geometry of the Circle NotesPowerPoint Presentation

Co-ordinate Geometry of the Circle Notes

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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

2θ

- 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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