10.5 Circles in the Coordinate Plane

Write equations and graph circles in the coordinate plane

1. Circles in the Coordinate Plane
The student is able to (I can):
• Write equations and graph circles in the coordinat e plane
• Use the equation and graph of a circle to solve pr oblems.

2. The equation of a circle is based on the Distance F ormula.
This means the radius can be found by
(h, k)
(x, y)
r
( ) ( )
2 2
2 1 2 1
d x x y y
= − + −
( ) ( )
2 2
r
x
h
y
k
= − + −
(
)
(
)
2 2
2
x h y k r
− + − =
Squaring both sides to
eliminate the square root
gives us the standard form of
a circle.

3. Examples
Identify the center and radius of the circles.
1.
2.
3.
(
)
(
)
2 2
7 2 16
x y
− + − =
(
)
(
)
2 2
5 1 36
x y
+ + − =
(
)
2
2
3 20
x y
+ + =

4. Examples
Identify the center and radius of the circles.
1.
center: (7, 2); radius:
2.
center: (–5, 1); radius:
3.
center: (0, –3); radius:
(
)
(
)
2 2
7 2 16
x y
− + − =
16 4
=
(
)
(
)
2 2
5 1 36
x y
+ + − =
36 6
=
(
)
2
2
3 20
x y
+ + =
20 2 5
=

5. To graph a circle once you know its center and radi us:
• Plot its center
• Count the radius from the center up, down, left, a nd right.
• Draw a smooth curve connecting the four points.
You can also use a compass. Plot the center as bef ore, open
up the compass to the length of the radius, and the n draw
the circle.

6. Examples
Write the equation of each circle.
1. ⊙Awith center A(4, –2)and radius 3
2. ⊙Bwith center B(–6, 3)that passes through (–2, 6)

7. Examples
Write the equation of each circle.
1. ⊙Awith center A(4, –2)and radius 3
2. ⊙Bwith center B(–6, 3)that passes through (–2, 6)
( ) ( )
(
)
2 2
2
4 2 3
x y
− + − − =
h k (
)
(
)
2 2
4 2 9
x y
− + + =
h k
x y
( )
(
)
( )
22
2
2 6
6 3
r
−
− + − =
−
2
25
r
=
(
)
(
)
2 2
6 3 25
x y
+ + − =