find out the intersection between a straight line and a curve parametrized by phi on a complex plane

Question

The equation for a straight line is simply: $$ y = mx + c $$ Let's say for now that the other curve is a unit circle $z = e^{i\phi}$, we know that: $$ y^2 + x^2 = 1 $$ which can simply be solved by substitution. Now consider the following: $$ f(z, \phi) = a(\phi)z^2 + bz + c = 0 $$ where the coefficient $a(\phi) = e^{i\phi}$, b and c are complex constants. There are 2 solutions for the polynomial root $z$: $$ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ which is implicitly a function of $\phi$ and should form a closed loop like how a circle would. My question is that, for a curve like this, I do not have a definition like the equation of a circle, so how should I go about solving the intersection between such a curve and a straight line?