Intermediate Algebra (11th edition)

Completing the Square to Find the Center and Radius

Find the center and radius of the circle and graph it.

Since the equation has - and -terms with equal coefficients, its graph might be

that of a circle. To find the center and radius, complete the squares on xand y.

Center-radius form

The final equation shows that the graph is a circle with center at and radius 5,

as shown in FIGURE 12.

1 - 1, - 32

3 x- 1 - 1242 + 3 y- 1 - 3242 = 52

1 x+ 122 + 1 y+ 322 = 25

1 x^2 + 2 x+ 12 + 1 y^2 + 6 y+ 92 = 15 + 1 + 9

c

1

2

16 2d

2

c = 9

1

2

12 2d

2

= 1

1 x^2 + 2 x 2 + 1 y^2 + 6 y 2 = 15

x^2 +y^2 + 2 x+ 6 y= 15

x^2 y^2

x^2 +y^2 + 2 x+ 6 y- 15 =0,

EXAMPLE 4

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

Transform so that the constant is on the right. Write in anticipation of completing the square. Square half the coefficient of each middle term. Complete the squares on both xand y. Factor on the left. Add on the right.

Add 1 and 9 on bothsides of the equation.

x

y

(–1, –3)^5

0

–5 3

x^2 + y^2 + 2x + 6y – 15 = 0 FIGURE 12 NOW TRY

NOTE Consider the following.

1. If the procedure of Example 4leads to an equation of the form

then the graph is the single point

2. If the constant on the right side is negative,then the equation has no graph.

OBJECTIVE 3 Recognize an equation of an

ellipse. An ellipseis the set of all points in a plane

the sumof whose distances from two fixed points is

constant. These fixed points are called foci(singu-

lar: focus). The ellipse in FIGURE 13has foci

and with x-intercepts and

and y-intercepts and It is shown in

more advanced courses that for an

ellipse of this type. The origin is the centerof the

ellipse.

An ellipse has the following equation.

c^2 =a^2 - b^2

1 0, b 2 1 0, -b 2.

1 - c, 0 2 , 1 a, 0 2 1 - a, 0 2

1 c, 0 2

1 h, k 2.

1 x-h 22 + 1 y-k 22 = 0 ,

x

y

(–c, 0)^0

Focus

Center

Ellipse