Question

Use a graphing utility to approximate the points of intersection of the graphs of the polar equations. Confirm your results analytically. $$ \begin{aligned} r &=3(1-\cos \theta) \\ r &=\frac{6}{1-\cos \theta} \end{aligned} $$

Short answer

The points of intersection of the two polar equations are (0,0) and (6, \pi).

Step-by-step solution

Graph the functions

Using a graphing utility, plot the two polar functions \( r = 3(1 - cos(\theta)) \) and \( r = \frac{6}{1 - cos(\theta)} \). Observe that the graphs intersect at two points.

Set the functions equal to each other

To find the precise points of intersection, set the two functions equal to each other and solve for \( \theta \):\n3(1 - cos(\theta)) = \frac{6}{1 - cos(\theta)}. Cross-multiplication yields: \(3 - 3cos(\theta) = 6 - 6cos(\theta)\). Solving for \( \theta \), we get \( \theta = 0 \) or \( \theta = \pi \).

Substitute values

Now, substitute \( \theta = 0 \) or \( \theta = \pi \) into the original equations to get the \( r \) values. \nFor \( \theta = 0 \), both equations yield \( r = 0 \) and for \( \theta = \pi \), both equations yield \( r = 6 \).