Conic Sections: Circles
Example 1
Write an Equation Given the Center and Radius
Write an equation for the circle with center at (5, –3) and radius 11.
(x – h)2 + (y – k)2 = r2
(x – 5)2 + [y – (–3)]2 = 112
(x – 5)2 + (y + 3)2 = 121
Equation of a circle
(h, k) = (5, –3), r = 11
Simplify.
The equation is (x – 5)2 + (y + 3)2 = 121.
Example 2
Write an Equation Given a Diameter
Write an equation for a circle if the endpoints of a diameter are at (–3, –7) and (2, 2).
First, find the center of the circle.
x1
(h, k) =
x2
2
–3
=
2
y1
,
,
y2
Midpoint Formula
2
–7
2
2
(x1, y1) = (–3, –7), (x2, y2) = (2, 2)
2
1
5
2
2
= – , –
Simplify.
Now find the radius.
( x2 – x1 )
r=
=
–
1
5
( y2 – y1 )
2
– (–3)
2
=
2
2
2
9
–
2
Distance Formula
2
– (–7 )
(x1, y1) = (–3, –7), (x2, y2) =
–
1
2
,–
5
2
2
Subtract.
2
106
=
Simplify.
4
106
The radius of the circle is
1
x
2
5
2
2
5
+ y
2
2
=
4
53
2
.
units, so r2 =
106
4
or
53
2
. An equation of the circle is
Example 3
Graph an Equation in Standard Form
Find the center and radius of the circle with equation (x – 1)2 + (y + 3)2 = 196. Then graph the circle.
Rewrite the equation as (x – 1)2 + [y – (–3)]2 = 142.
The center of the circle is (1, –3) and the radius is
14.
The table lists some integer values for x and y that
satisfy the equation.
x
1
1
15
–13
y
11
–17
–3
–3
Graph all of these points and draw the circle that passes through them.
Example 4
Graph an Equation not in Standard Form
Find the center and radius of the circle with equation x2 + y2 + 2x – 4y – 11 = 0. Then graph the
circle.
Complete the square.
x2 + y2 + 2x – 4y – 11 = 0
x2 + 2x + • + y2 – 4y + • = 11 + • + •
x2 + 2x + 1 + y2 – 4y + 4 = 11 + 1 + 4
(x + 1)2 + (y – 2)2 = 16
2
2
(x + 1) + (y - 2) = 16
The center of the circle is at (–1, 2), and the radius is 4. Locate the center and then find several points
located 4 units from the center. Draw the circle that passes through them.