Find the standard-form equation for a hyperbola which satisfies the given conditions.
-A hyperbola centered at the origin having focus at $( 9,0 )$ and eccentricity equal to $\frac { 3 } { 2 }$
A) $\frac { y ^ { 2 } } { 36 } + \frac { x ^ { 2 } } { 45 } = 1$
B) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 45 } = 1$
C) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 45 } = 1$
D) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 45 } = 1$

Question

Quizplus · Accepted Answer

The standard form of a hyperbola centered at the origin with a horizontal transverse axis is $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $2a$ is the length of the transverse axis, and $2b$ is the length of the conjugate axis. The distance from the center to a focus is $c$, and for this hyperbola, $c = 9$. The eccentricity $e = \frac{c}{a}$, and given $e = \frac{3}{2}$, we can find $a$ by rearranging the equation to $a = \frac{c}{e} = \frac{9}{\frac{3}{2}} = 6$. Thus, $a^2 = 36$. Since $c^2 = a^2 + b^2$, we find $b^2 = c^2 - a^2 = 81 - 36 = 45$. Therefore, the correct equation is $\frac{x^2}{36} - \frac{y^2}{45} = 1$.

Find the Standard-Form Equation for a Hyperbola Which Satisfies the Given