Question

Polar Equations to Rectangular Equations

Convert the polar equation to rectangular coordinates

$\sec \theta =2$

Short answer

The rectangular equation is$2x=\sqrt{x^2+y^2}$

Step-by-step solution

Step 1. Given information

An equation is given in polar coordinate system as$\sec \theta =2$

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,$\sec \theta =2$

$\begin{array}{l}\frac{1}{\cos \theta }=2\\ \Rightarrow 2\cos \theta =1\\ \Rightarrow 2r\cos \theta =r\\ \Rightarrow 2x=r\\ \Rightarrow 2x=\sqrt{x^2+y^2}\end{array}$