Chapter 10.3, Problem 26E

To determine

To replace: the polar equation $r^2\sin \left(2\theta \right)=2$ by an equivalent Cartesian(rectangular) equation and identify or describe the graph without using a grapher.

Answer to Problem 26E

The cartesian equation for the polar equation $r^2\sin \left(2\theta \right)=2$ is $xy=1$ and it is an equation of a hyperbola.

Explanation of Solution

Given information:

The polar equation is $r^2\sin \left(2\theta \right)=2$ .

Formula used:

  $\sin 2\theta =2\sin \theta \cos \theta$

  $x=r\cos \theta$ and $y=r\sin \theta$ .

The formula to convert polar equation to cartesian equation: $r^2=x^2+y^2$ .

Calculation:

The given equation is $r^2\sin \left(2\theta \right)=2$ .

Substitute $2\sin \theta \cos \theta$ for $\sin 2\theta$ in $r^2\sin \left(2\theta \right)=2$ .

  $r^2\left(2\sin \theta \cos \theta \right)=2$

From $x=r\cos \theta$ , $\cos \theta =\frac{x}{r}$ .

Similarly, from $y=r\sin \theta$ , $\sin \theta =\frac{y}{r}$ .

Substitute $\frac{x}{r}$ for $\cos \theta$ and $\frac{y}{r}$ for $\sin \theta$ in $r^2\left(2\sin \theta \cos \theta \right)=2$ .

  $\begin{array}{l}{r}^2\left(2\left(\frac{y}{r}\right)\left(\frac{x}{r}\right)\right)=2\\ 2r^2\frac{xy}{r^2}=2\\ 2xy=2\\ xy=1\end{array}$

Therefore, the cartesian equation for the polar equation $r^2\sin \left(2\theta \right)=2$ is $xy=1$ .

  $xy=1$ is an equation of a hyperbola.