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10.6

Equations of Circles

GOAL

1

F

E

C

INDING

QUATIONS OF

IRCLES

What

you should learn

GOAL

Write the equation

1

You can write an equation of a circle in a coordinate plane

y

(x, y )

of a circle.

if you know its radius and the coordinates of its center.

r

Suppose the radius of a circle is

r

and the center is

k).

(h,

Use the equation

GOAL

2

(h, k )

Let (x, y) be any point on the circle. The distance between

of a circle and its graph to

solve problems.

(x, y) and

(h, k)

is r, so you can use the Distance Formula.

x

Why

2

2

you should learn it

h

k

r

(x º

)

+ ( y º

)

=

To solve real-life

Square both sides to find the

standard equation of a circle

with radius r and

problems, such as deter-

center (h, k).

mining cellular phone

2

2

2

coverage, as in

(x º h)

+ (y º k)

r

=

Standard equation of a circle:

Exs. 41 and 42.

2

2

2

If the center is the origin, then the standard equation is x

+ y

= r

.

Writing a Standard Equation of a Circle

E X A M P L E 1

Write the standard equation of the circle with center

and radius 7.1.

(º4, 0)

S

OLUTION

2

2

2

(x º h)

+ (y º k)

=

r

Standard equation of a circle

2

2

2

[x º (º4)]

+ (y º 0)

=

7.1

Substitute.

2

2

(x + 4)

+ y

= 50.41

Simplify.

Writing a Standard Equation of a Circle

E X A M P L E 2

The point (1, 2) is on a circle whose center is (5, º1). Write the standard

equation of the circle.

S

OLUTION

the radius. The radius is the distance from the point (1, 2) to the center (5, º1).

Find

2

2

r = (5 º 1 )

+ ( º 1 º 2 )

Use the Distance Formula.

2

2

r = 4

+ ( º 3 )

Simplify.

r = 5

Simplify.

(h, k) =

and r =

into the standard equation of a circle.

Substitute

(5, º1)

5

2

2

2

(x º 5)

+ (y º (º1))

=

5

Standard equation of a circle

2

2

(x º 5)

+ (y + 1)

= 25

Simplify.

636

Chapter 10 Circles

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Parent category: Education