# Intersection of polar curve with line

I apologize for the horrible title. I came across this in an exam question: You're given $C_1$ as $$r = 1 + cos 2\theta$$ For $\frac{\pi}{2} \leq \theta \leq -\frac{\pi}{2}$. The only symmetry $C_1$ resembles is reflection about the polar axis.
It intersects with $C_2$, defined as $$r = 1 + sin \theta$$ , at the point $A$, which works out to be $(\frac{3}{2}, \frac{\pi}{6})$.

Then, a line is introduced, which is parallel to the polar axis, goes through $A$, and meets $C_1$ at a second point, $B$. Now you're supposed to show that the length of $OB$† equals some value, from which I only remember that it includes $\sqrt{13}$.
How is this solved? I.e. what is the exact length of $OB$ in surd form?

Intuitively, I tried to work out the angle by equating the distance from $A$ to the polar axis ($\frac{3}{4}$) with $r \sin \theta$, which is the vertical component of a polar point: $$(1 + \cos 2\theta)\sin \theta = \frac{3}{4}$$ Which can be transformed to $${\cos^2 \theta} ~\sin \theta = \frac{3}{8}$$ , which doesn't really yield anything simple - a depressed cubic in terms of $sin \theta$.

$O$ is the pole

• What is "the initial line"? There doesn't seem to be any other line defined ... – hmakholm left over Monica May 13 '15 at 22:39
• @HenningMakholm It seems I found the correct term: polar axis. – Columbo May 13 '15 at 22:44