OK, I wasn't on a class regarding this type of excercises. I got the notes from the lesson but have no idea how is it working. I hope you'll be able to clarify:
Determine the equation of the curve being a set of the middles of all the chords of the parabola $y=x^2-2$ going through (0,0). Sketch this curve.
let $y=mx$ the line through the origin, then we get $x^2-mx-2=0$ Solving this equation we obtain $x=\frac{m}{2}+\sqrt{\frac{m^2}{4}+2}$ $y=m(\frac{m}{2}+\sqrt{\frac{m^2}{4}+2})$ and $x=\frac{m}{2}-\sqrt{\frac{m^2}{4}+2}$ $y=m(\frac{m}{2}-\sqrt{\frac{m^2}{4}+2})$ thus the middle point of the chords is given by $M(m;m^2)$ and this equation is $y=x^2$.
..and all seems to be quite obvious but I have not the slightest idea why the middle moint is given by $M(m;m^2)$. How do the values of x'es an y'es we obtained imply that the middles of x and y are such and that the equation is y=$x^2$? Could you please help?