Stones Dropped into a Calm Pond at Points $A$ And $B$ Create Ripples That Propagate in Widening Circles

Stones dropped into a calm pond at points $A$ and $B$ create ripples that propagate in widening circles. In the figure below, points $A$ and $B$ are 26 feet apart, and the radii of the circles differ by 10 feet. The point $P ( x , y )$ where the circles intersect moves along a curve. Find the equation of the curve.

A)

$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 144 } = 1$$
B)
$$\frac { x ^ { 2 } } { 27 } + \frac { y ^ { 2 } } { 288 } = 1$$
C) $\quad \frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 144 } = 1$
D)
$$\frac { ( x - 10 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 144 } = 1$$