QuestionQuestion 5, 8.2.71
Write an equation for the hyperbola that has foci at $(0, \sqrt{5})$ and $(0,-\sqrt{5})$, and asymptotes $y= \pm 3 x$.
An equation for the hyperbola is $\square$ (Simplify your answer. Type your answer in standard form. Use integers or fractions for any numbers in the equation.)

Studdy Solution
Write the equation of the hyperbola in standard form:
Substitute $a^2$ and $b^2$ into the standard form of the vertical hyperbola:
$\frac{y^2}{\frac{9}{2}} - \frac{x^2}{\frac{1}{2}} = 1$
Multiply through by 2 to simplify:
$\frac{2y^2}{9} - 2x^2 = 1$
Multiply through by 9 to eliminate fractions:
$2y^2 - 9x^2 = 9$
The equation of the hyperbola is:
$\boxed{2y^2 - 9x^2 = 9}$