You've cleverly eliminated the answer choices from under the question, apparently to make it more interesting and more challenging for us.
If I understand your ponderous description of the equation, it looks like this:
(x+8)² + (y+9)² = 169 .
This is, in fact, the equation of a circle.
Its center is located at the point (-8, -9), and its radius is √169 = 13 .
