Find the Center and Radius x^2+y^2-4x+10y+13=0

Problem

x^2+y^2−4*x+10*y+13=0

Solution

1. Group the terms by variable and move the constant term to the right side of the equation.

(x^2−4*x)+(y^2+10*y)=−13

2. Complete the square for the x terms by adding ((−4)/2)^2=4 to both sides.

(x^2−4*x+4)+(y^2+10*y)=−13+4

3. Complete the square for the y terms by adding (10/2)^2=25 to both sides.

(x^2−4*x+4)+(y^2+10*y+25)=−13+4+25

4. Factor the trinomials into squared binomials and simplify the constant on the right side.

(x−2)^2+(y+5)^2=16

5. Identify the center and radius by comparing the result to the standard form (x−h)^2+(y−k)^2=r^2

h=2

k=−5

r^2=16⇒r=4

Final Answer

Center: *(2,−5), Radius: *4

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