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azatkgz

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## Homework Statement

Prove that the two families of parabolas

[tex]y^2=4a(a-x),a>0[/tex] and [tex]y^2=4b(b+x),b>0[/tex]

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.

## The Attempt at a Solution

Their tangent spaces at point [tex](x_0,y_0)[/tex] are

[tex]2y_0(y-y_0)+4a(x-x_0)=0[/tex]

[tex]2y_0(y-y_0)-4b(x-x_0)=0[/tex]

If they are perpendicular then we have

[tex]4y_0^2-16ab=0\Rightarrow y_0^2=4ab[/tex]

from the equations of parabolas we have

[tex]y_0^2=4a(a-x_0)[/tex]

[tex]y_0^2=4b(b+x_0)[/tex]

if we substitute [tex]x_0[/tex]

[tex]y_0^2=4ab[/tex]

So they are perpendicular.