Consider the system of equations, $$x^2+y^2+2gx+2fy+c=0,$$$$y^2=4ax.$$ How to show it is impossible to obtain more than 4 distinct real values for $x$ by solving them? If there are 4 distinct real values for $x$, then show that their sum is equal to zero.
I noticed these are two equations of a circle and a parabola. But, i could not find any satisfactory algabraic method. How to do this?
If you replace $x$ by $\frac{y^2}{4a}$ you get a polynomial in the fourth degree in $y$, so you can't have more than 4 solutions for $y$, which automatically means that you can't have more than 4 solutions for $x$.