If $y=-x^2+2x+4$ intersects $y=mx$ at $P$ & $ Q$. Determine the value $m$ so the mid point of $P$ & $Q$ is origin

Question

If $y=-x^2+2x+4$ intersects $y=mx$ at Point $P$ and Point $Q$. Determine the value m so the midpoint of P and Q is the origin.

I solved this question when I graphed it out, I wonder if there is a way to solve the problem without graphing?

Thank you.

Answer 1

Given $$y=-x^2+2x+4$$ Putting $y=mx$ in this equation to get the point of intersection, $$mx=-x^2+2x+4$$ $$\implies -x^2+(2-m)x+4=0$$ The roots of this equation will be the $x$ coordinates of $P(x_1,y_1),Q(x_2,y_2)$ as the midpoint is origin $$\frac{x_1+x_2}{2}=0$$ or sum of roots is $0$, $$\implies 2-m=0, \text{ or } m=2$$