Question

In Problems 30 and 31, convert each polar equation to a rectangular equation.

$r=\frac{4}{1-\cos \theta }$

Short answer

The rectangular equation is$y^2=8x+16$

Step-by-step solution

Step 1. Given Information

Given to convert the below polar equation to a rectangular equation.

$r=\frac{4}{1-\cos \theta }$

Step 2. Calculation

A polar equation is converted to rectangular equation by using the substitutions $r^2=x^2+y^2,r\cos \theta =x,r\sin \theta =y$

Rewriting the given equation to get the above terms:

$$\begin{gathered} r=\frac{4}{1-\cos \theta } \\ r\left(1-\cos \theta \right)=4 \\ r-r\cos \theta =4 \\ r=4+r\cos \theta \\ {r}^2={\left(4+r\cos \theta \right)}^2 \end{gathered}$$

Plugging the values:

$$\begin{gathered} x^2+y^2={\left(4+x\right)}^2 \\ {x}^2+y^2=16+8x+x^2 \\ {y}^2=8x+16 \end{gathered}$$

Step 3. Conclusion

The rectangular equation of the given polar equation is$y^2=8x+16$