Question

In Problems$11-18$, find an equation of the conic described. Graph the equation by hand.

Hyperbola; center at$(0,0)$; focus at$(0,4)$; vertex at $(0,-2)$

Short answer

The equation of the hyperbola is$\frac{y^2}{16}-\frac{x^2}{12}=1$

Step-by-step solution

Step 1. Given Information

Center at $(0,0)$$(0,0)$

Focus at $(0,4)$

Vertex at $(0,-2)$

Step 2. Calculation

Here, $a=4$

and $-ae=-2$

Therefore,$e=\frac{1}{2}$

Step 3. Calculation

Now, $a^2(1-e^2)=4^2(1-\frac{1}{4})$

$=12$

Step 4. Final Answer

The equation of the hyperbola is$\frac{y^2}{16}-\frac{x^2}{12}=1$

Step 4. Graph