It Can Be Shown That a Hyperbola with Center at the Origin

Multiple Choice

It can be shown that a hyperbola with center at the origin, foci at $\mathrm { F } ^ { \prime } ( - \mathrm { c } , 0 )$ and $\mathrm { F } ( \mathrm { c } , 0 )$ , and equation $d \left( P , F ^ { \prime } \right) - d ( P , F ) = 2 a$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ . Use this result to find an equation of a hyperbola with center at the origin, foci at $( - 8,0 )$ and $( 8,0 )$ , and absolute value of the distances from any point of the hyperbola to the two foci equal to 8 .

A) $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 8 } = 1$
B) $x ^ { 2 } - y ^ { 2 } = 8$
C) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 1$
D) $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 80 } = 1$