Find an equation in standard form for the hyperbola that satisfies the given conditions.
-Foci at $( \pm 9,0 )$, transverse axis with length 6
A) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 72 } = 1$
B) $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 81 } = 1$
D) $\frac { x ^ { 2 } } { 72 } - \frac { y ^ { 2 } } { 9 } = 1$

Question

Quizplus · Accepted Answer

The standard form of a hyperbola 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 the foci are at $(\pm c, 0)$. Given the foci at $(\pm 9,0)$ and the transverse axis length of 6 (implying $a = 3$), we find $c = 9$. Using the relationship $c^2 = a^2 + b^2$, we get $81 = 9 + b^2$, leading to $b^2 = 72$. Thus, the correct equation is $\frac{x^2}{9} - \frac{y^2}{72} = 1$.

Find an Equation in Standard Form for the Hyperbola That (±9,0)( \pm 9,0 )(±9,0)

Find an Equation in Standard Form for the Hyperbola That $( \pm 9,0 )$