Intermediate Algebra (11th edition)

To graph the circle, plot the center then

move 5 units right, left, up, and down from the

center, plotting the points

and

Draw a smooth curve through these four points,

sketching one quarter of the circle at a time. See

FIGURE 11.

1 9, - 32 , 1 - 1, - 32 , 1 4, 2 2 , 1 4, - 82.

1 4, - 32 ,

SECTION 11.2 The Circle and the Ellipse 643

0 x

y

( 44 , , – 3 )

(xx – 44 )^22 (yy 33 )^222525

5 5

8

6

FIGURE 11

Using the Center-Radius Form of the Equation of a Circle

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

Center-radius form

Let and

Simplify; NOW TRY

1 x+ 122 + 1 y- 222 = 7 A 2 aB^2 =a

3 x- 1 - 1242 + 1 y- 222 = A (^27) B h=-1,k=2, r= 27.
2

1 x- h 22 + 1 y-k 22 =r^2

1 - 1, 2 2 27

EXAMPLE 3

Equation of a Circle (Center-Radius Form)

An equation of a circle with radius rand center is

1 xh 22 1 yk 22 r^2.

1 h, k 2

Pay attention to signs here.

NOTE If a circle has its center at the origin then its equation becomes

Let in the center-radius form. See Example 1.

OBJECTIVE 2 Determine the center and radius of a circle given its equation.

In the equation found in Example 2,multiplying out and gives

Square each binomial. Subtract 25.

This general form suggests that an equation with both - and -terms with equal

coefficients may represent a circle.

x^2 y^2

x^2 + y^2 - 8 x+ 6 y=0.

x^2 - 8 x+ 16 + y^2 + 6 y+ 9 = 25

1 x- 422 + 1 y+ 322 = 25

1 x- 422 1 y+ 322

x^2 y^2 r^2.

1 x- 022 + 1 y- 022 =r^2 h=0, k= 0

1 0, 0 2 ,

NOW TRY
EXERCISE 2
Find an equation of the circle
with center at and
radius 3, and graph it.

1 - 2, 2 2

NOW TRY ANSWERS
2.

1 x+ 522 + 1 y- 422 = 6

x

y

–5^05

(–2, 2) 5

1 x+ 222 + 1 y- 222 = 9

NOW TRY
EXERCISE 3
Find an equation of the circle
with center at and

radius. 26

1 - 5, 4 2

NOW TRY

Examples 1 and 2suggest the form of an equation of a circle with radius rand

center at If is a point on the circle, then the distance from the center

to the point is r. By the distance formula,

Squaring both sides gives the center-radius formof the equation of a circle.

21 x-h 22 + 1 y-k 22 = r.

1 h, k 2 1 x, y 2

1 h, k 2. 1 x, y 2