Question

Convert each polar equation into a rectangular equation.

$r=\frac{1}{1+\cos \left(\theta \right)}$

Short answer

The rectangular equation is

$y^2+2x=1$

Step-by-step solution

Step 1. Given information

The polar equation is

$r=\frac{1}{1+\cos \left(\theta \right)}$

Step 2. Convert into the polar equation

The polar equation is

$r=\frac{1}{1+\cos \left(\theta \right)}$

Multiply denominator both sides

$$\begin{gathered} r\times (1+\cos (\theta \left)\right)=\frac{1}{1+\cos \left(\theta \right)}\times (1+\cos (\theta \left)\right) \\ r\times (1+\cos (\theta \left)\right)=1 \\ r+r\cos \left(\theta \right)=1 \end{gathered}$$

Apply conversion equation formula

$$\begin{gathered} r=\sqrt{x^2+y^2} \\ x=r\cos \left(\theta \right) \end{gathered}$$

So, the rectangular equation is

$\sqrt{x^2+y^2}+x=1$

Step 3. Remove square root

The equation is

$\sqrt{x^2+y^2}+x=1$

Subtract both sides $x$

localid="1647242304062" $$\begin{gathered} \sqrt{x^2+y^2}+x-x=1-x \\ \sqrt{x^2+y^2}=1-x \end{gathered}$$

Take square on both sides

localid="1647242312000" $$\begin{gathered} {\left(\sqrt{x^2+y^2}\right)}^2={(1-x)}^2{x}^2+y^2=1-2x+x^2 \\ {x}^2+y^2+2x-x^2=1 \\ {y}^2+2x=1 \end{gathered}$$