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) $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 80 } = 1$

Question

Quizplus · Accepted Answer

The given hyperbola has foci at $(-8, 0)$ and $(8, 0)$, so $c = 8$. The absolute value of the distances from any point on the hyperbola to the two foci is equal to $2a$, which is given as $8$, so $a = 4$. The relationship between $a$, $b$, and $c$ for a hyperbola is $c^2 = a^2 + b^2$. Substituting $a = 4$ and $c = 8$ gives $b^2 = 64 - 16 = 48$. Thus, the equation of the hyperbola is $\frac{x^2}{16} - \frac{y^2}{48} = 1$.

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