Moderator note:

The coordinates in the solution are switched.

Suppose that the point $(X, Y)$ is the center of a circle of radius $r$ which intersects the axis as given. Then, the y-intercepts would satisfy $(x-X) ^2 + (y-Y)^2 = r^2$ and $x = 0$ , or that $( y - Y)^2 = r^2 - X^2$ . The y-intercepts will be $y = Y \pm \sqrt{ r^2 - X^2}$ . Since the difference is 7, this tells us that $7 = 2 \sqrt{ r^2 - X^2}$ . Similarly, we get that $14 = 2 \sqrt{ r^2 - Y^2}$ , and thus the center of the circle satisfies $4 ( r^2 - Y^2) - 4 (r^2 - X^2) = 14^2 - 7^2$ , or that $4X^2 - 4Y^2 - 147 = 0$ .