Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

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

Short answer

The rectangular equation is$x^2+2y=1$

Step-by-step solution

Step 1. Given information

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

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

Multiply by$(1+\sin \theta )$on both sides,

$\begin{array}{l}\Rightarrow r(1+\sin \theta )=1\\ \Rightarrow r+r\sin \theta =1\end{array}$

Formulae used,

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

Substitute the values in the formulae,

$\begin{array}{l}r\sin \theta =1-r\\ r=1-r\sin \theta \\ r=1-y\end{array}$

Squaring on both sides,

$\begin{array}{l}\Rightarrow r^2\text{Invalid <m:msup> element}=\\ \text{Invalid <m:msup> element}\because =a^2-2ab+b^2\\ \Rightarrow r^2=1-2y+y^2\\ \Rightarrow x^2+y^2=1-2y+y^2\\ \Rightarrow x^2=1-2y\\ \Rightarrow x^2+2y=1\end{array}$