Equation of a circle

The idea

A circle with centre $(a, b)$ and radius $r$ is $(x - a)^2 + (y - b)^2 = r^2$. Complete the square in $x$ and in $y$ to find the centre and radius. A tangent is perpendicular to the radius at the point of contact.

1. Group the $x$ terms and the $y$ terms, then complete the square for each.
2. Move the constants to the right to get $r^2$.
3. For a tangent, find the radius gradient and take its negative reciprocal.

Watch out $(x + 4)^2$ means the centre has $x = -4$: the signs flip.