I have a seemingly easy question, but I have no clue how to find out its answer. I have the function
$$f(x)=\tfrac{1}{8} x^2$$
This function is for (a parabolic cross-section through) a paraboloid and I am searching for the point where, when this point emits light, every lightwave is parallel after reflection in the paraboloid.
I know the solution is the point $P(0;2)$ but I have no clue why. The problem I have is that I have no way of using derivatives for this, otherwise it would be fairly easy. I hope you can help me…
Go in reverse. If you have the directrix at $y=-f$ and focus at $(x_f,y_f)=(0,f)$, then you can write down the definition of parabola:
$$||(x,y)-(0,f)||=y-(-f)$$ Square this: $$x^2+(y-f)^2=(y+f)^2$$ $$x^2=4yf$$ $$y=\frac{x^2}{4f}$$