Equations Of Circles Worksheet - Chapter 10-8
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6Equations 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
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