Answer
The equation for the hyperbola described in the problem is:
$\frac{y^{2}}{8^{2}} - \frac{x^{2}}{6^{2}} = 1$
Work Step by Step
With vertices at $(0, ±8)$, we know that the hyperbola has a vertical transverse axis. We now plug in $8$ for $a$ to get:
$\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$
$\frac{y^{2}}{8^{2}} - \frac{x^{2}}{b^{2}} = 1$
Now we simplify the equation to get:
$\frac{y^{2}}{64} - \frac{x^{2}}{b^{2}} = 1$
By using the value we have for $a$, we can come up with the value for $b$ by using the following equation:
$a^{2} + b^{2} = c^{2}$
$a^{2} + b^{2} = c^{2}$
We get the value for $c$ from the foci. In this case, $c$ is $10$. We can plug this value into the equation as well:
$8^{2} + b^{2} = 10^{2}$
Simplify to get:
$64 + b^{2} = 100$
Subtract $64$ from both sides to get:
$b^{2} = 36$
Take the square root of both sides to get:
$b = 6$
Now that we have both values for $a$ and $b$, we plug these into the standard equation for hyperbolas to get:
$\frac{y^{2}}{8^{2}} - \frac{x^{2}}{6^{2}} = 1$
