Question

Find the equation of the circle in standard form:

Answer 1

(x + 5)^2 + (y + 1)^2 = 2^2

Explanation:

I see that the instructions here call for "completing the square."

We need to rewrite x^2 + y^2 + 10x + 2y + 22 = 0 in the standard equation-of-a-circle formula (x - h)^2 + (y - k)^2 = r^2.

Start with x^2 + y^2 + 10x + 2y + 22 = 0, Group x terms together, then y terms:

x^2 + 10x + y^2 + 2y = -22

Going through the steps of completing the square, we insert additional constants:

x^2 + 10x + 25 - 25 + y^2 + 2y + 1 - 1 = -22, or

x^2 + 10x + 25 + y^2 + 2y + 1 = + 25 + 1 - 22 = 4

Rewrite x^2 + 10x + 25 as (x + 5)^2, and y^2 + 2y + 1 as (y + 1)^2

after which the original equation becomes (x + 5)^2 + (y + 1)^2 = 2^2