Q.381. Angle between tangents drawn from a point to a parabola.

Question 381.
What is the angle between the tangents drawn from the point (1,4) to the parabola y^2 = 4x ?

Answer 381.
y = mx + a/m is the tangent to the parabola y^2 = 4ax
For y^2 = 4x, a = 1
=> y = mx + 1/m is the tangent

If it passes through (1, 4),
4 = m + 1/m
=> m^2 - 4m + 1 = 0
The roots m1 and m2 are the slopes of the tangents
=> m1 + m2 = 4 and m1m2 = 1
=> m1 - m2
= √[(m1 + m2)^2 - 4m1m2]
= √(16 - 4)
= 2√3

If θ = angle between the tangents, then
tanθ
= l (m1 - m2) / (1 + m1m2) l
= l 2√3 / (1 + 1) l
= √3
=> θ = arctan√3 = π/3.

Link to YA!