Tangent plane.

[azatkgz]

Please,help me with this problem.
1. The problem statement, all variables and given/known data
Prove that the two families of parabolas
y^2=4a(a-x),a>0 and y^2=4b(b+x),b>0
form an orthogonal net. Specifically, check that for any a, b > 0 these two parabolas
are perpendicular to each other at the points where they intersect.

3. The attempt at a solution

Their tangent spaces at point (x_0,y_0) are

2y_0(y-y_0)+4a(x-x_0)=0

2y_0(y-y_0)-4b(x-x_0)=0

If they are perpendicular then we have

4y_0^2-16ab=0\Rightarrow y_0^2=4ab

from the equations of parabolas we have

y_0^2=4a(a-x_0)

y_0^2=4b(b+x_0)

if we substitute x_0

y_0^2=4ab
So they are perpendicular.