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 $y^2=4x$
Here, $4a=4\Rightarrow a=1$
Equation of tangent at $(x,y)$ is
$y=mx+\frac{a}{m}$
Here, $(x,y)=(1,4)$
$\therefore 4=m+\frac{1}{m}$
$\Rightarrow m^2-4m+1=0$
$\Rightarrow m=\frac{4\pm \sqrt{16-4}}{2}=\frac{4\pm 2\sqrt{3}}{2}=2\pm \sqrt{3}$
Let $m_1=2+\sqrt{3},m_2=2-\sqrt{3}$
Let angle between the tangents is ${\theta }_1$ Then,
$\tan \theta =\frac{m_1-m_2}{1+m_1{m}_2}$
$=\frac{2+\sqrt{3}-2+\sqrt{3}}{1+(2+\sqrt{3})(2-\sqrt{3})}$
$=\frac{2\sqrt{3}}{1+4-3}$
$\Rightarrow \tan \theta =\sqrt{3}=\tan \frac{\pi }{3}$
$\therefore \theta =\frac{\pi }{3}$