# Graph x^2+y^2-10x+2y-23=0

x2+y2-10x+2y-23=0
Add 23 to both sides of the equation.
x2+y2-10x+2y=23
Complete the square for x2-10x.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=-10,c=0
Consider the vertex form of a parabola.
a(x+d)2+e
Substitute the values of a and b into the formula d=b2a.
d=-102(1)
Simplify the right side.
Cancel the common factor of 10 and 2.
Factor 2 out of 10.
d=-2⋅52⋅1
Cancel the common factors.
Factor 2 out of 2⋅1.
d=-2⋅52(1)
Cancel the common factor.
d=-2⋅52⋅1
Rewrite the expression.
d=-51
Divide 5 by 1.
d=-1⋅5
d=-1⋅5
d=-1⋅5
Multiply -1 by 5.
d=-5
d=-5
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise -10 to the power of 2.
e=0-1004⋅1
Multiply 4 by 1.
e=0-1004
Divide 100 by 4.
e=0-1⋅25
Multiply -1 by 25.
e=0-25
e=0-25
Subtract 25 from 0.
e=-25
e=-25
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x-5)2-25
(x-5)2-25
Substitute (x-5)2-25 for x2-10x in the equation x2+y2-10x+2y=23.
(x-5)2-25+y2+2y=23
Move -25 to the right side of the equation by adding 25 to both sides.
(x-5)2+y2+2y=23+25
Complete the square for y2+2y.
Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=2,c=0
Consider the vertex form of a parabola.
a(x+d)2+e
Substitute the values of a and b into the formula d=b2a.
d=22(1)
Cancel the common factor of 2.
Cancel the common factor.
d=22⋅1
Divide 1 by 1.
d=1
d=1
Find the value of e using the formula e=c-b24a.
Simplify each term.
Raise 2 to the power of 2.
e=0-44⋅1
Multiply 4 by 1.
e=0-44
Divide 4 by 4.
e=0-1⋅1
Multiply -1 by 1.
e=0-1
e=0-1
Subtract 1 from 0.
e=-1
e=-1
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(y+1)2-1
(y+1)2-1
Substitute (y+1)2-1 for y2+2y in the equation x2+y2-10x+2y=23.
(x-5)2+(y+1)2-1=23+25
Move -1 to the right side of the equation by adding 1 to both sides.
(x-5)2+(y+1)2=23+25+1
Simplify 23+25+1.
(x-5)2+(y+1)2=48+1
(x-5)2+(y+1)2=49
(x-5)2+(y+1)2=49
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=r2
Match the values in this circle to those of the standard form. The variable r represents the radius of the circle, h represents the x-offset from the origin, and k represents the y-offset from origin.
r=7
h=5
k=-1
The center of the circle is found at (h,k).
Center: (5,-1)
These values represent the important values for graphing and analyzing a circle.
Center: (5,-1)