Finding the equation of a hyperbola given the vertices and foci.

Question

A hyperbola has the vertices $(0,0)$ and $(0,-16)$ and the foci $(0,2)$ and $(0,-18)$. Find the equation with the given information.

Answer 1

Your vertices and foci lie on the y axis. This means that your hyperbola opens upward.

Equation for a generic hyperbola that opens upward:

$-\frac{(x-h)^2}{a^2} + \frac{(y-v)^2}{b^2} = 1$

The center $(h,v)$ lies at the mid-point between the two foci.

$a$ = distance from center to each vertex.

$c^2 = a^2+b^2$, Where c is the distance from the center to the focus.

Answer 2

The hyperbola is opening up in direction $y:$

$$ a = 8;\, b = 6; c = \sqrt {a^2 + b^2 } = a\,e =10\,;$$

Carefully choose sign to shift axis downward so the top vertex touches x-axis.

$$ \dfrac{(y+a)^2}{a^2} - \dfrac {x^2}{b^2} =1. $$