Question

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

• A. $0$
• B. $\frac{\mathrm{\pi }}{6}$
• C. $\frac{\mathrm{\pi }}{4}$
• D. $\frac{\mathrm{\pi }}{3}$

Answer 1

The correct option is D

$\frac{\mathrm{\pi }}{3}$

Explanation for the correct option:

We know that equation of a tangent of parabola, whose slope is $m$ is given by $y=mx+\frac{1}{m}$

Since it passes through $\left(1,4\right)$, so

$$\begin{gathered} 4=m+\frac{1}{m} \\ \Rightarrow m^2-4m+1=0 \end{gathered}$$

solving above quadratic equation, we get

$m_1=2+\sqrt{3},m_2=2-\sqrt{3}$

Now, we have

$\left|m_1-m_2\right|=2\sqrt{3},m_1\times m_2=1$

Therefore,

$$\begin{gathered} \tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right| \\ \Rightarrow \tan \theta =\frac{2\sqrt{3}}{2} \\ \Rightarrow \tan \theta =\sqrt{3} \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$

Hence, the correct option is (D)