Circles

Objectives: •Write an equation for a circle given sufficient information•Given an equation of a circle, graph it and label the radius and the center

Standard Equation of a Circle

OP = r

P(x,y)

Or

2 2(x 0) (y 0) r

2 2x y r

2 2 2x y r

Standard Equation of a Circle

2 2 2x y r

An equation for the circle with its center at (0,0) and a radius of r is

Example 1Write the standard equation of the circle whose center is at the origin and whose radius is 4. Sketch the graph.

2 2 2x y r 2 2 2x y 4

2 2x y 16 -4 -2

2

42

4

-4

-2

Standard Equation of a Circle

The standard equation for a translated circle is(x – h)2 + (y – k)2 = r2

center: (h, k)radius: r

-6-8

6

8

2

4

-4 -2

Example 2Write the standard equation of the circle graphed below.

2 2 2(x h) (y k) r 2 2 2(x ( 2)) (y 3) 4

2 2(x 2) (y 3) 16

Practice

1) C(0,0) radius: 9

Write the standard equation of a circle with the following center and radius.

2) C(2,3) radius: 53) C(-5,2) radius: 4

Practice

1) x2 + y2 = 25

Graph each equation. Label the center and radius.

2) (x – 2)2 + y2 = 43) (x + 4)2 + (y – 3)2 = 49

Center = (0, 0) radius = 5

Center = (2, 0) radius = 2

Center = (-4, 3) radius = 7

Example 3Write the standard equation for the circle given by x2 + y2 – 12x – 2y - 8 = 0. State the coordinates of its center and give its radius. 2 2x y 12x 2y 8 0

2 2x 12x y 2y 8 2 2(x 12x ) ( 1 136 y ) 32y 68

2 2(x 6) (y 1) 45

Center: (6,1)Radius: 45 3 5

Example 4Write the standard equation for the circle given by x2 + y2 + 6x – 4y - 3 = 0. State the coordinates of its center and give its radius. Then sketch the graph.

2 2x y 6x 4y 3 0 2 2x 6x y 4y 3

2 2(x 6x ) ( 4 4y ) 39 4y 9 2 2(x 3) (y 2) 16

Center: (-3,2)Radius: 16 4

-6-8

6

8

2

4

-4 -2

PracticeWrite the standard equation for the circle given by x2 + y2 - 2x + 2y - 7 = 0. State the coordinates of its center and give its radius. Then sketch the graph.

9)1()1( 22 yx

Center = (1, 1)

Radius = 3