CIRCLES

Unit 3-2

Equations we’ll need:

Distance formula

Midpoint formula

2 2

1 2 1 2d x x y y

1 2 1 2( , ) ,2 2

x x y yM x y

Definitions

Circle: Set of all points in a plane that is a fixed distance from a fixed point

Center: The fixed equidistance point of the circle

Radius: The fixed equidistance length of the circle

Circle Equation

2 2 2( ) ( )x h y k r

( , )Center h k

Radius r

Example 1

Name the center and the radius of this circle,

and graph it2 2( 2) ( 3) 16x y

2 2 2( ) ( )x h y k r ( 2,3)

4

c

r

2 16

4

r

r

Example 1

Name the center and the radius of this circle, and graph it

2 2 2( 2) ( 3) 4x y

2 2( 2) ( 3) 16x y

( 2,3)

4

c

r

Example 2Write an equation of the circle with its center at

(–3, 5), the radius is 5 and graph it.

2 2 2( ) ( )x h y k r

Example 3Name the center and the radius of this circle,

and graph it

2 2 2( ) ( )x h y k r

2 2 25x y

YOUR TURN!Graph the equation 2 2( 3) ( 2) 9x y

2.3 - Circles 1004/18/23 19:36

Example 4

Find the radius, graph the circle, and write the equation in standard form with the given center of (2, -3) and the point (1, 0).

2 2 2( ) ( )x h y k r

C

radius

2 2

1 2 1 2d x x y y

2 22 1 3 0d

2 21 3d

1 9 10d radius

2.3 - Circles 1104/18/23 19:36

Example 4

Find the radius, graph the circle, and write the equation in standard form with the given center of (2, -3) and the point (1, 0).

2 2 2( ) ( )x h y k r

C

radius =

10

10

(2, 3)

r

C

22 2( 2) ( ( 3)) 10x y

2 2( 2) ( 3) 10x y

2.3 - Circles 1204/18/23 19:36

Write a circle equation whose center is at the origin and passes through (1, –6).

2 2 2( ) ( )x h y k r Example 5

2.3 - Circles 1304/18/23 19:36

Write a circle equation whose center is at (1, 0) and passes through (-2, 3)

2 2 2( ) ( )x h y k r YOUR TURN!

2.3 - Circles 1404/18/23 19:36

Write a circle equation whose endpoints of a diameter are (-5, 2) and (3, 6).

1 2 1 2,2 2

x x y yMP

2 2 2( ) ( )x h y k r Example 6

Diameter

5 3 2 6,

2 2MP

1,4MP Center

2.3 - Circles 1504/18/23 19:36

Write a circle equation whose endpoints of a diameter are (-5, 2) and (3, 6).

2 2 2( ) ( )x h y k r Example 6

2 2

1 2 1 2d x x y y

1,4MP Center

Center

Radius

2 25 ( 1) 2 4d

2 24 2d

16 4 20d Radius

2.3 - Circles 1604/18/23 19:36

Write a circle equation whose endpoints of a diameter are (-5, 2) and (3, 6).

2 2 2( ) ( )x h y k r Example 6

20

( 1,4)

r

C

2 2( 1) ( 4) 20x y

22 2( ( 1)) ( 4) 20x y

2.3 - Circles 1704/18/23 19:36

YOUR TURN!

Write a circle equation whose endpoints of a diameter are (5, 4) and (-1, -6).

1 2 1 2,2 2

x x y yMP

2 2 2( ) ( )x h y k r

2 2

1 2 1 2d x x y y

2.3 - Circles 1804/18/23 19:36

Determine the equation of this circle by the graph given

Example 7

2.3 - Circles 1904/18/23 19:36

YOUR TURN!

Determine the equation of this circle by the graph given

2.3 - Circles 2004/18/23 19:36

Assignment

WORKSHEET!