Write the equation of the hyperbola with foci at $(0,15)$ and $(0,-15)$ and the transverse axis of length 20.
A) $4 y^{2}-5 x^{2}=500$
B) $5 y^{2}-4 x^{2}=500$
C) $13 y^{2}-4 x^{2}=1300$
D) $4 y^{2}-13 x^{2}=1300$

Question

Quizplus · Accepted Answer

The foci are on the y-axis, so the hyperbola is vertical. The distance between the foci is 30, so $2c = 30$ giving $c = 15$. The transverse axis length is 20, so $2a = 20$ giving $a = 10$. Using the relationship $c^2 = a^2 + b^2$, we find $b^2 = 125$. The equation is $\frac{y^2}{100} - \frac{x^2}{125} = 1$, which simplifies to $5y^2 - 4x^2 = 500$.

Write the Equation of the Hyperbola with Foci At (0,15)(0,15)(0,15) And (0,−15)(0,-15)(0,−15)

Write the Equation of the Hyperbola with Foci At $(0,15)$ And $(0,-15)$