Solve the problem.
-Find the equation of the hyperbola whose vertices are (0, 10)and (0, -10)and whose asymptotes arere $y = \frac { 5 } { 6 } x$ an and $y = - \frac { 5 } { 6 } x$
A) $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 144 } = 1$
B) $\frac { y ^ { 2 } } { 144 } - \frac { x ^ { 2 } } { 100 } = 1$
C) $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 25 } = 1$
D) $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 36 } = 1$

Question

Quizplus · Accepted Answer

The vertices at (0, 10) and (0, -10) indicate that the hyperbola is vertical, with its center at the origin (0,0) and a vertical distance of 10 units from the center to each vertex. This gives $a^2 = 100$. The slopes of the asymptotes $\pm \frac{5}{6}$ are derived from the equation of the hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, where the slope of the asymptotes is $\pm \frac{a}{b}$. Thus, $\frac{a}{b} = \frac{5}{6}$, and since $a^2 = 100$, solving for $b^2$ gives $b^2 = 144$. Therefore, the equation of the hyperbola is $\frac{y^2}{100} - \frac{x^2}{144} = 1$.

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