The equation of any tangent to y2=4ax is
If it touches x2=4ay then, the equation.
⇒x2=4a(mx+ma) has equal roots.
⇒mx2−4am2x−4a2=0 has equal roots
If the roots are equal, the discriminant is equal to 0
Putting m=−1 in y=mx+am we get,
⇒x+y+a=0 This is the common tangent to the two parabola.