College Algebra, Chapter 8, 8.3, Section 8.3, Problem 22

Find the equation for the hyperbola whose graph is shown below.







The hyperbola $\displaystyle \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ has a vertical transverse axis, its vertices $(0, \pm a)$ and foci $(0, \pm c)$ are located on the $y$-axis. Notice from the graph that the hyperbola have vertices on $(0, \pm 12)$ and foci $(0, \pm 13)$.

Recall that $c^2 = a^2 + b^2$, so


$
\begin{equation}
\begin{aligned}

b^2 =& c^2 - a^2
\\
\\
b =& \sqrt{c^2 - a^2}
\\
\\
b =& \sqrt{13^2 - 12^2}
\\
\\
b =& 5

\end{aligned}
\end{equation}
$


Therefore, the equation is


$
\begin{equation}
\begin{aligned}

& \frac{y^2}{12^2} - \frac{x^2}{5^2} = 1
\\
\\
& \text{or}
\\
\\
& \frac{y^2}{144} - \frac{x^2}{25} = 1

\end{aligned}
\end{equation}
$