Problem on co-ordinate geometry

Question

Suppose the circle with equation $x^2 + y^2 + 2fx + 2gy + c = 0$ cuts the parabola $y^2 = 4ax$, ($a > 0$) at four distinct points. If d denotes the sum of ordinates of these four points, then find the set of possible values of d.

Answer 1

As Mark Bennet has pointed out, if you substitute x by $\frac{y^2}{4a}$ in equation $x^{2}+y^{2}+2fx+2gy+c=0$ , you get $y^{4}+0.y^{3}+4a(1+2f)y^{2}+32a^{2}gy+16a^{2}c=0$. By Vieta's formula sum of the roots of the above equation is = - $\frac{\textrm{coefficient of } y^3}{\textrm{coefficiet of }y^4} =0 $. Answer is 0.