Answer
The equation for the hyperbola specified is:
$y^{2} - \frac{x^{2}}{3} = 1$
Work Step by Step
The vertices for this hyperbola are $(0, ±1)$. This means that this hyperbola has a vertical axis. We can plug in $1$ for $a$ into the standard equation to get:
$\frac{x^{y}}{x^{y}} - \frac{x^{y}}{x^{y}} = 1$
$\frac{y^{2}}{1^{2}} - \frac{x^{2}}{b^{2}} = 1$
We also have the value for $c$ by looking at the foci given. the foci are $(0, ±2)$.
Now we plug in the values for $a$ and $c$ into the following equation:
$a^{2} + b^{2} = c^{2}$
$1^{2} + b^{2} = 2^{2}$
Simplify to get:
$1 + b^{2} = 4$
Subtract $1$ from both sides to get:
$b^{2} = 3$
Take the square root of both sides to isolate $b$:
$b = \sqrt 3$
Now that we have values for $a$ and $b$, we plug these values into the standard equation for hyperbolas to get:
$\frac{y^{2}}{1^{2}} - \frac{x^{2}}{(\sqrt 3)^{2}} = 1$
We simplify to get:
$y^{2} - \frac{x^{2}}{3} = 1$
