10.7 Write and Graph Equations of Circles

HubarthGeometry

In the circle below, let point (x, y) represent any point on the circle whose centeris at the origin. Let r represent the radius of the circle.

In the right triangle,

r = length of hypotenusex = length of a legy = length of a leg

By the Pythagorean Theorem, you can write

x2 + y2 = r2

This is an equation of a circle with center at the origin.

r

x

y

Ex 1 Write an Equation of a CircleWrite an equation of the circle.

Solution

The radius is 4 and the center is at the origin.

2 2 2x y r 2 2 2

2 2

4

16

x y

x y

Standard Equation of a Circle If the center of a circle is not at the origin, you can use the Distance Formula to write an equation of the circle.

For example, the circle shown at the right has center (3, 5) and a radius of 4.

Let (x, y) represent any point on the circle.Use the Distance Formula to find the lengthsof the legs.

leg: I x-3 I leg: I y-5 Ihypotenuse: 4

Use these expressions in the Pythagorean Theorem to find an equationof the circle.

(x-3)2 + (y-5)2=42

This is an example of the standard equation of a circle.

.(3, 5)

(x, y)

4I y-5 I

I x-3 I

Standard Equation of a Circle

In the coordinate plane, the standard equation of a circle with center at (h, k) and radius r is

(x-h)2 + (y-k)2 = r2

x-coordinate ofthe center

y-coordinate ofthe center

.r

(x, y)

(h, k)

Ex 2 Write the Standard Equation of a Circle.Write the standard equation of the circle with center (2, -1)and radius 3.

.(2, -1)

Solution2 2 2

2 2 2

( ) ( )

( 2) ( ( 1)) 3

x h y k r

x y

2 2( 2) ( 1) 9x y

Ex 3 Graph a CircleGraph the given equation of the circle.

2 2a. (x-1) ( 2) 4y 2 2b. (x+2) 4y

Solution

a. The center is (1, 2) and the radius is 2

.

b. The center is (-2, 0) and the radius is 2

.(1, 2) (-2, 0)

PracticeWrite an equation of the circle.

1. 2.

2 23. (x-1) 16y

.

2 2 4x y 2 2 25x y