Intermediate Algebra (11th edition)

A circle may not be centered at the origin, as seen in the next example.

Finding an Equation of a Circle and Graphing It

Find an equation of the circle with center at and radius 5, and graph it.

1 x- 422 + 1 y+ 322 = 25 Square each side.

21 x- 422 + 3 y- 1 - 3242 = 5

1 4, - 32

EXAMPLE 2

642 CHAPTER 11 Nonlinear Functions, Conic Sections, and Nonlinear Systems

OBJECTIVES When an infinite cone is intersected by a plane, the resulting figure is called a conic

section.The parabola is one example of a conic section. Circles, ellipses, and hyper-

bolas may also result. See FIGURE 9.

The Circle and the Ellipse

11.2

1 Find an equation

of a circle given the

center and radius.

2 Determine the

center and radius

of a circle given

its equation.

3 Recognize an

equation of

an ellipse.

4 Graph ellipses.

Circle

Ellipse Parabola Hyperbola

FIGURE 9

OBJECTIVE 1 Find an equation of a circle given the center and radius.A

circleis the set of all points in a plane that lie a fixed distance from a fixed point. The

fixed point is called the center,and the fixed distance is called the radius.We use the

distance formula fromSection 8.3to find an equation of a circle.

Finding an Equation of a Circle and Graphing It

Find an equation of the circle with radius 3 and center at and graph it.

If the point is on the circle, then the distance

from to the center is 3.

Distance formula

Let

and

Square each side.

An equation of this circle is The graph is

shown in FIGURE 10.

x^2 + y^2 = 9.

x^2 +y^2 = 9

d=3.

21 x- 022 + 1 y- 022 = 3 x 1 =0, y 1 =0,

21 x 2 - x 122 + 1 y 2 - y 122 = d

1 x, y 2 1 0, 0 2

1 x, y 2

1 0, 0 2 ,

EXAMPLE 1

3

0 3 x

y

3

(x, x, y)

xx^22 yy^2299

FIGURE 10

NOW TRY

NOW TRY
EXERCISE 1
Find an equation of the circle
with radius 6 and center at
1 0, 0 2 , and graph it.

NOW TRY ANSWER

x

y

(^06)
6
x^2 +y^2 = 36
Let and
in the distance formula.
x 1 =4,y 1 =-3, d= 5