Math 135
Circles and Completing the Square
Solutions
Using the method of completing the square, put each circle into the form
(x − h)2 + (y − k)2 = r2
.
Then determine the center and radius of each circle.
1. x2 + y 2 − 10x + 2y + 17 = 0.
Answer 1.
x2 + y 2 − 10x + 2y + 17
(x2 − 10x) + (y 2 + 2y)
(x2 − 10x + 25) + (y 2 + 2y + 1)
(x − 5)2 + (y + 1)2
=
=
=
=
0
−17
−17 + 25 + 1
32
=
=
=
=
0
−16
−16 + 16 + 9
32
Circle of radius r = 3 centered at (5, −1).
2. x2 + y 2 + 8x − 6y + 16 = 0.
Answer 2.
x2 + y 2 + 8x − 6y + 16
(x2 + 8x) + (y 2 − 6y)
(x2 + 8x + 16) + (y 2 − 6y + 9)
(x + 4)2 + (y − 3)2
Circle of radius r = 3 centered at (−4, 3).
3. 9x2 + 54x + 9y 2 − 18y + 64 = 0.
Answer 3.
9x2 + 54x + 9y 2 − 18y + 64
(9x2 + 54x) + (9y 2 − 18y)
9(x2 + 6x) + 9(y 2 − 2y)
9(x2 + 6x + 9) + 9(y 2 − 2y + 1)
9(x + 3)2 + 9(y − 1)2
(x + 3)2 + (y − 1)2
=
=
=
=
=
0
−64
−64
−64 + 81 + 9
26
√ !2
26
=
3
√
Circle of radius r =
University of Hawai‘i at Mānoa
26
3
centered at (−3, 1).
57
R Spring - 2014
Math 135
Circles and Completing the Square
Solutions
4. 4x2 − 4x + 4y 2 − 59 = 0.
Answer 4.
4x2 − 4x + 4y 2 − 59
(4x2 − 4x) + (4y 2 )
4(x2 − x) + 4(y 2 )
1
4(x2 − x + ) + 4(y 2 )
4
1 2
4(x − ) + 4(y 2 )
2
1
(x − )2 + (y 2 )
2
Circle of radius
University of Hawai‘i at Mānoa
√
= 0
= 59
= 59
= 59 + 1
= 60
= 15
15 centered at ( 12 , 0).
58
R Spring - 2014