Prove that the algebraic sum of the ordinates of intersection of a circle and a parabola is 0

Question

I consider two curves $x^2=4ay$ and $x^2+y^2=\lambda^2$

So $$y^2+4ay-\lambda^2=0$$ And $$y_1+y_2\not =0$$

I just want to whether the question itself is right or not

Answer 1

For $x^2=4ay$, sum of abscissae will be zero due to symmetry about y-axis.

For $y^2=4ax$, sum of ordinates will be zero due to symmetry about x-axis.

(Perhaps the question has been mixed up? Or was it originally a true/false question?)

Answer 2

The equation $y^2+4ay-\lambda^2=0$ has the solutions

$$y_{1/2}=-2a \pm \sqrt{4a^2+\lambda^2}.$$

Hence

$$y_1+y_2=-4a.$$