Chapter 8.1, Problem 64E

To determine

To convert the given polar equation to rectangular coordinates

  $r=\frac{2}{1-\cos \theta }$

Answer to Problem 64E

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

Explanation of Solution

Given: Polar equation: $r=\frac{2}{1-\cos \theta }$

Formula Used:

A polar equation is any equation that describes a relation between r and θ , where r represents the distance from the pole (origin) to a point on a curve, and θ represents the counter-clockwise angle made by a point on a curve, the pole, and the positive x-axis.

Also, $x=r\cos \theta$ and $y=r\sin \theta$

  $r=\sqrt{x^2+y^2}$ and $\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$

Calculation:

Given Polar equation is $r=\frac{2}{1-\cos \theta }$

Using the above definition, we have:

  $r=\frac{2}{1-\cos \theta }$ ;

Solving further, we have:

  $r\left(1-\cos \theta \right)=2$

  $r-\left(r\cos \theta \right)=2$

  $\sqrt{x^2+y^2}-x=2$

Thus, the rectangular equation is $\sqrt{x^2+y^2}-x=2$ .

Conclusion:

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