Question

Two hyperbolas that have the same set of asymptotes are called conjugates. Show that the hyperbolas

$\frac{x^2}{4}-y^2=1\text{and}{y}^2-\frac{x^2}{4}=1$

are conjugate. Graph each hyperbola on the same set of coordinate axes.

Short answer

The graph of the hyperbola is

Step-by-step solution

Step 1. Given Information

Two hyperbolas that have the same set of asymptotes are called conjugates. Show that the hyperbolas $\frac{x^2}{4}-y^2=1\text{and}{y}^2-\frac{x^2}{4}=1$are conjugate.

Step 2. Graph of the hyperbola 

$\frac{x^2}{4}-y^2=1\text{and}{y}^2-\frac{x^2}{4}=1$

The asymptote of 1st hyperbola is

$$\begin{gathered} y=\pm \frac{b}{a}x \\ y=\pm \frac{1}{2}x \end{gathered}$$

The asymptote of 2nd hyperbola is

$$\begin{gathered} y=\pm \frac{a}{b}x \\ y=\pm \frac{1}{2}x \end{gathered}$$

The asymptotes of both the hyperbolas are the same.

The graph of the hyperbola is