What is the equation of a circle which touches both axes and passes through 4,-2?

Answer 1

The circle must lie in QIV.

The center of the circle is (a, -a) with a radius of a

(x - a)² + (y + a)² = a²

Since the circle passes through (4, -2), that pair must solve.

(4 - a)² + (-2 + a)² = a²

I leave it to you to solve for a and answer. (Note there are two possible circles).