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

Multiple Choice

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 12 feet apart, and the radii of the circles differ by 4 feet. The point $P ( x , y )$ where the circles intersect moves along a curve. Find the equation of the curve.

A) $\quad \frac { ( x - 5 ) ^ { 2 } } { 4 } - \frac { ( y - 4 ) ^ { 2 } } { 32 } = 1$
B) $\quad \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 32 } = 1$
C) $\quad \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 64 } = 1$
D) $\quad \frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 32 } = 1$
E) none of these