Equations of Circles 10.6

CA State Standards17: Prove theorems

using coordinate geometry.

Lesson GoalsWrite the equation of a circle.Graph a circle on the coordinate plane.

ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

2 2 2 is used to find lengths of a right triangle.c a b

Previously on Geometry

definitionEquation of a Circle

On the coordinate plane a circle is defined by (x – h)2 + (y – k)2 = r2

(h, k) is the coordinate of the centerr is the radius of the circle

(h, k)

(x, y)r

Give the center and radius of the circle

2 2) 4 2 25a x y

22) 1 17b x y

) c center 4, 2

radius 5

center 0, 1

radius 17 center 2,2

radius 4

2 21 65 1x y

2 2 25 1 4x y

2 2 2x y rh k

exampleWrite the equation of a

circle.Center: (5, 1) Radius: 4

2 2 22 14 3 r

22 24 3x y r

2 2 2x y rh k

exampleWrite the equation of a

circle.Center: (4, 3) Point on Circle: (2,1)

2 2 22 4 r 220 r

2 23 04 2x y

Graph the circle:example

Center =

Radius =

2 22 3 9x y

2, 33

Graph the circle:example

Center =

Radius =

2 23 1 25x y

3, 15

The equation of a circle is

Example: solution graphically 2 213 6 9x y

Are the following points on, inside or outside the circle?

BA

C

10, 6

14, 8

20, 9

A

B

C

8

on

inside

outside

solution algebraically (write in column)If you plug in a point (x, y) and

then (x, y) is outside the circle

then (x, y) is inside the circle

2 2 2x h y k r then (x, y) is on the circle

2 2 2x h y k r

2 2 2x h y k r

B

The equation of a circle is

Example: solution algebraically 2 213 6 9x y

2 210 13 6 6 ? 9

2_ 3 0 ? 9

9 9

2 214 13 8 6 ? 9

2 21 2 ? 9

5 9

2 220 13 9 6 ? 9

2 27 3 ? 9

58 9

A

C

10, 6 14, 8

20, 9

A B

C

Are the following points on, inside or outside the circle?

on

inside outside

SummaryExplain the equation of a circle.How can you tell algebraically if a point is inside, on, or outside a circle?

13

Today’s Assignment

Tribe Pride

always my best effort.

p. 638: 8 – 40 e

Give the center and radius of the circle.

Give the coordinates of the center, the radius, and the equation of the circle.

Write the standard equation of the circle with the given center and radius.

Use the given information to write the standard equation of the circle.

Graph the equation.

The rest at home