What is the equation of common tangent to the parabola \(y^2\) = 4ax and \(x^2\) = 4ay ?

Solution :

The equation of tangent in slope form to \(y^2\) = 4ax is

y = mx + \(a\over m\)

Now, if it is common to both parabola, it also lies on second parabola

then \(x^2\) = 4a(mx + \(a\over m\))

\(mx^2 – 4am^2 – 4a^2\) = 0 has equal roots.

then its discriminant is zero. i.e. \(b^2-4ac\) = 0

\(16a^2m^2 + 16a^2m\) = 0

m = -1

Putting m = -1 in equation y = mx + \(a\over m\) we get

y = -x – a

x + y + a = 0

Which is the required equation of common tangent to both parabola.