Find the standard-form equation of the hyperbola centered at the origin which satisfies the given conditions.
-Foci at $( 0 , - 5 ) , ( 0,5 )$; asymptotes: $y = \frac { 4 } { 3 } x , y = - \frac { 4 } { 3 } x$
A) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$
B) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 9 } = 1$
C) $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$
D) $\frac { y ^ { 2 } } { 9 } - \frac { x ^ { 2 } } { 16 } = 1$

Question

Quizplus · Accepted Answer

The foci being on the y-axis indicates a vertical transverse axis. The distance between the center and a focus (c) is 5, and the slope of the asymptotes ($b/a$) is $4/3$, implying $a^2 + b^2 = c^2$ and $b/a = 4/3$. Solving these gives $a^2 = 16$ and $b^2 = 9$, leading to the equation $\frac{y^2}{16} - \frac{x^2}{9} = 1$.

Find the Standard-Form Equation of the Hyperbola Centered at the Origin