Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$r=\frac{4}{1+2\sin \theta }$

Short answer

The rectangular equation is$x^2+y^2={(4-2y)}^2$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$r=\frac{4}{1+2\sin \theta }$

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,$r=\frac{4}{1+2\sin \theta }$

Multiply by r on both sides,

$\begin{array}{l}{r}^2=\frac{4r}{1+2\sin \theta }\\ \Rightarrow r^2=\frac{4r^2}{r+2r\sin \theta }\\ \Rightarrow x^2+y^2=\frac{4(x^2+y^2)}{r+2y}\\ \Rightarrow r+2y=\frac{4(x^2+y^2)}{(x^2+y^2)}\\ \Rightarrow r+2y=4\\ \Rightarrow r=4-2y\\ \Rightarrow r^2\text{Invalid <m:msup> element}=\\ \Rightarrow x^2+\text{Invalid <m:msup> element}{y}^2=\end{array}$