The hint that's given is what makes this problem doable - completing the square. Completing the square asks you to find an appropriate number so that you make a perfect square trinomial, like (x + 1)².
We are told the circle's equation is x² + y² + 4x - 12y + 4 = 0. Let's first put the 4 on the other side so we don't have to deal with it.
x² + y² + 4x - 12y + 4 = 0 original equation
x² + y² + 4x - 12y = -4 subtracting 4 on both sides
x² + 4x + y² - 12y = -4 rearrange the terms so x are together, y are together
x² + 4x + ___ + y² - 12y + = -4 Add spots for completing the square
Let's fill in the first blank. You want something that makes a perfect square. If you multiply (a + b) and (a + b), you would get a² + 2ab + b². So your middle term is double the end terms, and the last term is the square of it. What happens is you take the coefficient - 4 - and to halve it and square it to find the term for completing the square. Half of 4 is 2, and squaring 2 gives you 4. The number in the middle - 2 - completes the square. A check by FOIL shows that (x + 2)² = x² + 2x + 2x + 4 = x² + 4x + 4.
x² + 4x + 4 + y² - 12y + = -4 +4 Add 4 to both sides and complete the square
(x + 2)² + y² -12y + = 0 Simplify the right side, factor the left.
Let's repeat the process on the y terms using halving and squaring. Half of 12 is 6, its square is 36, and the go-between is 6. The squared number, like before, gets added.
(x + 2)² + y² -12y + 36 = 0 + 36 Add 36 to both sides
(x + 2)² + (y - 6)² = 36 Factor the left, simplify the right
In the form above we have put the circle into center-radius form, (x - a)² + (y - b)² = r² where (a, b) is the center and r is the radius. The signs on the original formula are negative, so when we have a positive, it means we need its negative. That makes the x part as -2. The signs match on y, so we take the 6 for the y part. The radius is the square root of r². √36 = 6 (we throw out the negative since it's a length).
A) Thus the center of the circle is at (-2, 6)
B) And the radius is 6.
