Chapter 11.3, Problem 26E

To determine

To determine: The equivalent Cartesian equation for the given polar equation. Also, identify the graph without using a grapher.

Answer to Problem 26E

The required equation is $xy=1$ and the equation is an equation of a rectangular hyperbola.

Explanation of Solution

Given information:

The polar equation: $r^2\sin 2\theta =2$

Calculation:

The given equation is $r^2\sin 2\theta =2$ .

Rewrite the given equation as:

  $\begin{array}{c}{r}^2\sin 2\theta =2\\ r^2\left(2\sin \theta \cos \theta \right)=2\\ \frac{\cancel{2}\left(r\cos \theta \right)\left(r\sin \theta \right)}{\cancel{2}}=\frac{\cancel{2}}{\cancel{2}}\\ \left(r\cos \theta \right)\left(r\sin \theta \right)=1\end{array}$

To find the required Cartesian equation, use the formulas $x=r\cos \theta$ , $y=r\sin \theta$ , $r^2=x^2+y^2$ , $\sin 2\theta =2\sin \theta \cos \theta$ and $\tan \theta =\frac{y}{x}$ .

Substitute $r\cos \theta$ for $x$ and $r\sin \theta$ for $y$ in the equation $\left(r\cos \theta \right)\left(r\sin \theta \right)=1$ .

  $\begin{array}{c}\left(r\cos \theta \right)\left(r\sin \theta \right)=1\\ xy=1\end{array}$

The above equation is an equation of a rectangular hyperbola.

Hence, the required equation is $xy=1$ and the equation is an equation of a rectangular hyperbola.