Question

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

Hyperbola; center at $(-2,-3)$; focus at $(-4,-3)$; vertex at $(-3,-3)$

Short answer

The equation of the hyperbola is$\frac{{(x+2)}^2}{1}-\frac{{(y+3)}^2}{3}=1$

Step-by-step solution

Step 1. Given Information

Center of hyperbola at $(-2,-3)$

The focus of hyperbola at $(-4,-3)$

The vertex of the hyperbola is at $(-3,-3)$

Step 2. Calculation

Here, $(\alpha ,\beta )=(-2,-3)$

and $(\alpha +ae,\beta )=(-4,-3)$

and$(\alpha +a,\beta )=(-3,-3)$

Step 3. Calculation

Equating both sides we get

$\alpha =-2,\beta =-3$

$a=-1,e=2$

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

$=3$

Then the equation of the hyperbola is

$\frac{{(x+2)}^2}{1}-\frac{{(y+3)}^2}{3}=1$

Step 4. Graph