Answer:
Subtract
31
31
from both sides of the equation.
x
2
+
y
2
−
4
x
−
12
y
=
−
31
x2+y2-4x-12y=-31
Complete the square for
x
2
−
4
x
x2-4x
.
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(
x
−
2
)
2
−
4
(x-2)2-4
Substitute
(
x
−
2
)
2
−
4
(x-2)2-4
for
x
2
−
4
x
x2-4x
in the equation
x
2
+
y
2
−
4
x
−
12
y
=
−
31
x2+y2-4x-12y=-31
.
(
x
−
2
)
2
−
4
+
y
2
−
12
y
=
−
31
(x-2)2-4+y2-12y=-31
Move
−
4
-4
to the right side of the equation by adding
4
4
to both sides.
(
x
−
2
)
2
+
y
2
−
12
y
=
−
31
+
4
(x-2)2+y2-12y=-31+4
Complete the square for
y
2
−
12
y
y2-12y
.
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(
y
−
6
)
2
−
36
(y-6)2-36
Substitute
(
y
−
6
)
2
−
36
(y-6)2-36
for
y
2
−
12
y
y2-12y
in the equation
x
2
+
y
2
−
4
x
−
12
y
=
−
31
x2+y2-4x-12y=-31
.
(
x
−
2
)
2
+
(
y
−
6
)
2
−
36
=
−
31
+
4
(x-2)2+(y-6)2-36=-31+4
Move
−
36
-36
to the right side of the equation by adding
36
36
to both sides.
(
x
−
2
)
2
+
(
y
−
6
)
2
=
−
31
+
4
+
36
(x-2)2+(y-6)2=-31+4+36
Simplify
−
31
+
4
+
36
-31+4+36
.
(
x
−
2
)
2
+
(
y
−
6
)
2
=
9
(x-2)2+(y-6)2=9
This is the form of a circle. Use this form to determine the center and radius of the circle.
(
x
−
h
)
2
+
(
y
−
k
)
2
=
r
2
(x-h)2+(y-k)2=r2
Match the values in this circle to those of the standard form. The variable
r
r
represents the radius of the circle,
h
h
represents the x-offset from the origin, and
k
k
represents the y-offset from origin.
r
=
3
r=3
h
=
2
h=2
k
=
6
k=6
The center of the circle is found at
(
h
,
k
)
(h,k)
.
Center:
(
2
,
6
)
(2,6)
These values represent the important values for graphing and analyzing a circle.
Center:
(
2
,
6
)
(2,6)
Radius:
3
3