Find the Center and Radius x^2+y^2+2x-4y+1=0

Problem

x^2+y^2+2*x−4*y+1=0

Solution

1. Group the x terms and y terms together and move the constant term to the right side of the equation.

(x^2+2*x)+(y^2−4*y)=−1

2. Complete the square for the x terms by adding (2/2)^2=1 to both sides.

(x^2+2*x+1)+(y^2−4*y)=−1+1

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

(x^2+2*x+1)+(y^2−4*y+4)=−1+1+4

4. Factor the quadratic expressions into perfect square binomials and simplify the right side.

(x+1)^2+(y−2)^2=4

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

h=−1

k=2

r^2=4⇒r=2

Final Answer

Center: *(−1,2), Radius: *2

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