Question

Two equal parabola have the same vertex and their axis are at right angle prove that their common tangent touch each at the end of their latus rectum

Answer 1

Let the parabolas be :

y² = 4ax

x² = 4ay

The tangent of the first parabola at a point (a, - 2a)

That is at the one end of the latus rectum is :

Y = x + a

Putting this value at the other equation we get it cuts the other parabola at :

(-2a, a).

That is at the extremity of the latus rectum.

Hence this is proved.