The angle between the tangents drawn from the point (1, 4) to the parabola `y^2=4x` is `pi/6` (b) `pi/4` (c) `pi/3` (d) `pi/2`

Options

A. `pi//6`
B. `pi//4`
C. `pi//3`
D. `pi//2`

Answer

C

Explanation

The equation of any tangents of slope m to the parabola `y^(2)=4x` is
`y=mx+1/m" …(i)"`
If it passes through P(1, 4), then
`4=-m=1/mrArrm^(2)-4m+2=0`
Let `m_(1), m_(2)=4" and "m_(1)m_(2)=1`
`m_(1)+m_(2)=4" and "m_(1)m_(2)=1`
Let `theta` be the angle between the tangents drawn from P. Then,
`tantheta=(m_(1)-m_(2))/(1+m_(1)m_(2))=(sqrt((m_(1)+m_(2))^(2)-4m_(1)m_(2)))/(1+m_(1)+m_(2))`
`rArr" "tantheta=sqrt(16-4)/2=sqrt3rArrtheta=pi/3`