Find condition on constants such that two parabolas are tangent

by RoRi

May 31, 2016

Let $a \neq 0$ , and consider the two parabolas
$y^2 = 4p(x-a), \qquad \text{and} \qquad x^2 = 4qy.$

If these two parabolas are tangent to each other, show that the $x$ -coordinate of the contact point depends only on $a$ .
Find conditions for the constants $a,p,q$ such that the two parabolas are tangent to each other.