y = mx + a/m is the tangent to the parabola y2 = 4ax
Let us take value of 'a' to be 1 here (such that answer is shown cleaner. Won't make any difference)
For y2 = 4x, a = 1
=> y = mx + 1/m is the tangent
If it passes through (1, 4),
4 = m + 1/m
=> m2 - 4m + 1 = 0
The roots m1 and m2 are the slopes of the tangents.
Use sum of roots and product of roots in a Quadratic Equation.
=> m1 + m2 = 4 and m1m2 = 1
Let us calculate value of m1 - m2
=> m1 - m2
= √[(m1 + m2)2 - 4m1m2]
= √(16 - 4)
If θ = angle between the tangents, then
= l (m1 - m2) / (1 + m1m2) l
= l 2√3 / (1 + 1) l
=> θ = tan-1√3 = π/3.