Question

The angle between the tangents drawn from the point $ (1, 4) $ to the parabola $ y^2 = 4x $ is

• A. $ \pi/2 $
• B. $ \pi/6 $
• C. $ \pi/4 $
• D. $ \pi/3 $

Answer 1

Answer: (D)

Given equation of parabola is $y^{2}=4 x$
Here, $ 4 a=4 \Rightarrow a=1$
Equation of tangent at $(x, y)$ is
$y=m x+\frac{a}{m} $
Here, $ (x, y)=(1,4)$
$\therefore 4=m+\frac{1}{m}$
$\Rightarrow m^{2}-4 m+1=0 $
$\Rightarrow m=\frac{4 \pm \sqrt{16-4}}{2}=\frac{4 \pm 2 \sqrt{3}}{2}=2 \pm \sqrt{3}$
Let $m_{1}=2+\sqrt{3}, m_{2}=2-\sqrt{3}$
Let angle between the tangents is $\theta_{1}$ Then,
$\tan \theta=\frac{m_{1}-m_{2}}{1+m_{1} m_{2}} $
$=\frac{2+\sqrt{3}-2+\sqrt{3}}{1+(2+\sqrt{3})(2-\sqrt{3})} $
$=\frac{2 \sqrt{3}}{1+4-3}$
$\Rightarrow \tan \theta=\sqrt{3}=\tan \frac{\pi}{3}$
$ \therefore \theta=\frac{\pi}{3}$