Question

Will polar curves always intersect?

Answer 1

No. Two curves need not intersect.

Every curve can be expressed in either polar or rectangular form. Some are simpler in one form than the other, but there are not two classes (or families) of curves.

The curves `x^2+y^2=1` and `x^2+y^2=9` are concentric circles with unequal radii. They do not intersect.

In polar form, these are the curves `r=1` and `r=3`. (And, of course, they still do not intersect.)