Co-ordinate Geometry of the Circle Notes

1 / 31

# 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 &amp; 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).

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Co-ordinate Geometry of the Circle Notes' - cerise

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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:(x – h)2 + (y – k)2 = r2

K

• r
• Method
• Centre: c(h, k)
• c

(c) Aidan Roche 2009

K

• r
• Method
• Centre: c(0, 0)
• c

(c) Aidan Roche 2009

To find centre and radius of K. Given 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

• 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

• 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?

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?

• 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 (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

(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

• 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

• 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: 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