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 Equation of a circle
(x – 5)2 + [y – (–3)]2 = 112 (h, k) = (5, –3), r = 11
(x – 5)2 + (y + 3)2 = 121 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.
(h, k) =
x1 x2
2, y1 y2
2
Midpoint Formula
=
–3 2
2, –7 2
2
(x1, y1) = (–3, –7), (x2, y2) = (2, 2)
=
–1
2, – 5
2
Simplify.
Now find the radius.
r =
(x2 – x1)2 (y2 – y1)2
Distance Formula
=
–1
2 – (–3)2 –5
2 – (–7)2
(x1, y1) = (–3, –7), (x2, y2) =
15
– , –
22
=
5
2
2 9
2
2
Subtract.
=
106
4
Simplify.
The radius of the circle is
106
4
units, so r2 =
106
4
or
53
2
. An equation of the circle is
2
1
2
x
+
2
5
2
y
=
53
2
.
