Polar Coordinates — Question 4

Question 4

Problem

Determine all intersections of $r=2$ and $r=4\cos\theta$.

Question 4 – Solution

Solution

1. Use the polar–Cartesian relationships $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$ Equivalent polar coordinates satisfy $$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).$$

2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

3. $2=4\cos\theta$ gives $\theta=\pm\pi/3$.

4. The points are $(1,\pm\sqrt3)$.