Question

Answer Problems 9–11 using the figure.

The equation of the hyperbola is of the form

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

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

Short answer

The equation of the hyperbola is of the form$\frac{{(y-h)}^2}{a^2}-\frac{{(x-k)}^2}{b^2}=1$since the graph of the hyperbola opens up and down.

Step-by-step solution

Step 1. Given Information

The graph of the hyperbola is:

We need to find if the hyperbola is of the form$\frac{{(x-h)}^2}{a^2}-\frac{{(y-k)}^2}{b^2}=1$or$\frac{{(y-h)}^2}{a^2}-\frac{{(x-k)}^2}{b^2}=1$

Step 2. Concept

If the equation of the hyperbola is of the form $\frac{{(y-k)}^2}{a^2}-\frac{{(x-h)}^2}{b^2}=1$, then the graph of the hyperbola opens up and down.

If the equation of the hyperbola is of the form $\frac{{(x-h)}^2}{a^2}+\frac{{(y-k)}^2}{b^2}=1$then the graph of the hyperbola opens towards the left and right.

Step 3. Explanation

Since the graph of the hyperbola opens up and down, the equation of the hyperbola is of the form$\frac{{(y-h)}^2}{a^2}-\frac{{(x-k)}^2}{b^2}=1$.