Question

In Problems 19–28, find an equation for the hyperbola described. Graph the equation by hand. Foci at $(-4,0)$and $(4,0)$; asymptote the line$y=-x$.

Short answer

Equation of the hyperbola:$\frac{x^2}{8}-\frac{y^2}{8}=1$

Graph of the hyperbola:

Step-by-step solution

Step 1. Given information 

Hyperbola with foci at $(-4,0)$and $(4,0)$; asymptote the line$y=-x$.

Step 2. Required to find 

Find the equation of the hyperbola. Graph the equation.

Step 3. Finding the equation of the hyperbola 

Given, hyperbola with foci at $(-4,0)$and $(4,0)$; asymptote the line $y=-x$.

Comparing with standard form of hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,b^2=c^2-a^2;\mathrm{foci}=(\pm c,0);\mathrm{asymptotes},\mathrm{y}=\pm \frac{b}{a}\mathrm{x}$

We get,

$$\begin{gathered} c=4,\frac{b}{a}=1 \\ {b}^2=c^2-a^2 \\ \Rightarrow b^2+a^2=c^2 \\ \Rightarrow b^2+b^2=c^2 \\ \Rightarrow 2b^2=c^2 \\ \Rightarrow 2b^2=4^2 \\ \Rightarrow b^2=\frac{16}{2} \\ \Rightarrow b^2=8 \\ \Rightarrow b=\pm 2\sqrt{2}=a \end{gathered}$$

Substituting $a$and $b$in the equation,

$\frac{x^2}{8}-\frac{y^2}{8}=1$

Step 4. Graphing the hyperbola 

Graph of the hyperbola: