Polar Cycling and Graph Intersections

One way to approach this problem is by graphing the two equations. Alex’s graph is a rose with 4 petals that surrounds the polar axis and the line $\theta = \frac{\pi}{2}$ , and Brenda’s graph is a cardioid that faces downward. By looking at the graphs we can clearly see they intersect 7 times, so the pair will collide 7 times, right?

Polar Graphs

The issue with this method is that the polar coordinate $(r, \theta)$ lies on the same point graphically as $(-r, \theta + \pi)$ . We can view this when we look at the starting point of our two bikers, Alex at $(1, 0)$ and Brenda at $(-1, 0)$ . The graphs intersect at the location $(1, 0)$ because Brenda’s graph contains the point $(-1, \pi)$ , but the bikers won’t be at that point at the same time. We need to exclusively find the number of coordinates that are contained in both equations, not the points of graphical intersection.

We can do this by setting the equations equal to other and doing some algebra, starting with

$\cos 2\theta = -\sin \theta -1$ ,

and then using the double angle formula and rearranging to rewrite as,

$2 \sin^2 \theta - \sin \theta - 2 = 0$ .

We can use the quadratic formula now to find that,

$\sin \theta = \frac{1 \pm \sqrt{17}}{4}$ ,

and since $\frac{1 + \sqrt{17}}{4}$ is outside the range of the $\sin$ function,

$\theta = \{ \arcsin (\frac{1 - \sqrt{17}}{4}), \pi - \arcsin (\frac{1 - \sqrt{17}}{4}) \}$ .

This leaves us with $\boxed{2}$ distinct values for theta where both riders will be the same distance from the origin and collide.