Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r^2=\sin 2\theta$

Short answer

The rectangular equation is${(x^2+y^2)}^2=2xy$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r^2=\sin 2\theta$

Step 2. Concept used

To convert rectangular coordinates to polar coordinates, use the formulae below-

$\begin{array}{l}r=\sqrt{x^2+y^2}\\ \theta ={\tan }^{-1}\left(\frac{y}{x}\right)\end{array}$

Convert polar coordinates using the same formulas-

$\begin{array}{l}x=r\cos \theta \\ y=r\sin \theta \end{array}$

Substitute the value in the formulae. And simplify them.

Step 3. Calculation

The polar equation is given as,$r^2=\sin 2\theta$

$\begin{array}{l}{r}^2=\sin 2\theta \\ r^2=2\sin \theta \cos \theta \end{array}$

By using the formulae above,

$\begin{array}{l}{r}^2=2\sin \theta \cos \theta \\ \Rightarrow r^2\times r\times r=2r\sin \theta \times r\cos \theta \\ \Rightarrow r^4=2xy\\ \Rightarrow r^2\times r^2=2xy\\ \Rightarrow {(x^2+y^2)}^2=2xy\end{array}$