Write an equation for the hyperbola.
-vertices at $( 0,4 ) , ( 0 , - 4 )$; foci at $( 0,10 ) , ( 0 , - 10 )$
A) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 84 } = 1$
B) $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 16 } = 1$
C) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 100 } = 1$
D) $\frac { y ^ { 2 } } { 84 } - \frac { x ^ { 2 } } { 16 } = 1$

Question

Quizplus · Accepted Answer

The equation of a hyperbola with 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. Here, $2a = 8$ (from $-4$ to $4$), so $a^2 = 16$, and $2c = 20$ (from $-10$ to $10$), so $c^2 = 100$. Using $c^2 = a^2 + b^2$, we find $b^2 = 100 - 16 = 84$. Therefore, the correct equation is $\frac{y^2}{16} - \frac{x^2}{84} = 1$.

Write an Equation for the Hyperbola (0,4),(0,−4)( 0,4 ) , ( 0 , - 4 )(0,4),(0,−4)

Write an Equation for the Hyperbola $( 0,4 ) , ( 0 , - 4 )$