Find the standard form of the equation of the hyperbola with the given characteristics. vertices: $( 0 , \pm 3 )$ foci: $( 0 , \pm 6 )$
A) $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 27 } = 1$
B) $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 36 } = 1$
C) $\frac { y ^ { 2 } } { 9 } + \frac { x ^ { 2 } } { 27 } = 1$
D) $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 27 } = 36$
E) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 27 } = 36$

Question

Quizplus · Accepted Answer

The standard form of a hyperbola with a vertical transverse axis is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where $2a$ is the distance between the vertices, and $2c$ is the distance between the foci. Given vertices at $(0, \pm 3)$, $a = 3$, so $a^2 = 9$. Given foci at $(0, \pm 6)$, $c = 6$, so $c^2 = 36$. Using $c^2 = a^2 + b^2$, we find $b^2 = 36 - 9 = 27$. Thus, the equation is $\frac{y^2}{9} - \frac{x^2}{27} = 1$.

Find the Standard Form of the Equation of the Hyperbola (0,±3)( 0 , \pm 3 )(0,±3)

Find the Standard Form of the Equation of the Hyperbola $( 0 , \pm 3 )$