How do you write an equation for the hyperbola with vertices (0,2) and (0,-2) and foci at (0,9) and (0,-9)?

May 11, 2016

${x}^{2} / 4 - {y}^{2} / 77 = 1$

Explanation:

Center C bisects the line joining the vertices $\left(0 , - 2\right) \mathmr{and} \left(0 , 2\right)$

So C is at the origin.

The distance between the vertices is the transverse axis 2a = 4.

So, a = 2.

The distance between the foci = 2a X eccentricity (e).

So, 4e = 18. e = 9/2.

The semi-transverse axis $b = a \sqrt{{e}^{2} - 1}$.

So, $b = 2 \sqrt{\frac{81}{4} - 1} = \sqrt{77}$

${a}^{2} = 4 \mathmr{and} {b}^{2} = 77$.

Now, the equation of the hyperbola takes the standard form

${x}^{2} / 4 - {y}^{2} / 77 = 1$..