Polar Coordinates — Question 8

Question 8

Problem

Which description better reveals the geometry of $x^2+y^2=5x$: Cartesian or polar? Convert it and explain.

Question 8 – 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. Polar form is $r=5\cos\theta$, immediately revealing a circle through the pole with diameter $5$ along the polar axis.