# Tangent plane.

1. Feb 5, 2008

### azatkgz

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.