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Equations of Circles

•

Write the equation of a circle.

•

Graph a circle on the coordinate plane.

kind of equations describes

kind of equations describes

the ripples of a splash?

the ripples of a splash?

When a rock enters the water, ripples move

out from the center forming concentric circles.

If the rock is assigned coordinates, each ripple

can be modeled by an equation of a circle.

EQUATION OF A CIRCLE

The fact that a circle is the locus of points in a plane

equidistant from a given point creates an equation for any circle.

Suppose the center is at (3, 2) and the radius is 4.

y

P

( x , y )

The radius is the distance from the center. Let P(x, y)

be the endpoint of any radius.

( 3, 2 )

2

2

d

(x

x

)

(y

y

)

Distance Formula

2

1

2

1

2

2

4

(x

3 )

(y

2)

4, (x

, y

)

(3, 2)

d

1

1

O

x

2

2

16

(x

3)

(y

2)

Square each side.

Applying this same procedure to an unknown center (h, k)

and radius r yields a general equation for any circle.

Standard Equation of a Circle

An equation for a circle with center

r

at (h, k) and radius of r units is

2

2

2

(x

h)

(y

k)

r

.

( h , k )

Study Tip

Equation of a Circle

Example

Example

1

1

Equation of

Write an equation for each circle.

Circles

a. center at ( 2, 4), d

4

Note that the equation of

If d

4, r

2.

a circle is kept in the form

shown above. The terms

2

2

2

(x

h)

(y

k)

r

Equation of a circle

being squared are not

2

2

2

[x

( 2)]

[y

4]

2

(h, k)

( 2, 4), r

2

expanded.

2

2

(x

2)

(y

4)

4

Simplify.

b. center at origin, r

3

2

2

2

(x

h)

(y

k)

r

Equation of a circle

2

2

2

(x

0)

(y

0)

3

(h, k)

(0, 0), r

3

2

2

x

y

9

Simplify.

Lesson 10-8 Equations of Circles 575

Pete Turner/Getty Images

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Parent category: Education