if y=[(tanx)..

The first parabola S₁ = x² - 4y =0
and the second parabola S₂ = y² - 4x =0
Consider the equation S₁ - S₂ = 0
That is
x² - y² +4(x-y) =0
=> (x-y) (x + y + 4) =0
Since for the common points x≥0, y≥0, so x+y+4=0 does not hold for the intersection point.
Accord x-y=0 is the common chord. If (h,k) be the intersection of tangents, then the chord of contact is hx = 2(y+k) i.e. h2x - y = k
But this equation must be the same as x-y = 0. So h = 2 and k=0. So the required intersection point is (2, 0).