Q95 / 160 · AP EAMCET 2019 (21 Apr Shift 1) · Mathematics
The angle between the tangents drawn to the parabola $y^2=4 x$ from the point $(1,4)$ is
- A$\frac{\pi}{4}$
- Correct answerB$\frac{\pi}{3}$
- C$\frac{2 \pi}{5}$
- D$\frac{\pi}{6}$
Solution
Equation of a tangent to a parabola $y^2=4 x$ is
$$
y=m x+\frac{1}{m}
$$
Since, it passes through point $(1,4)$. So,
or
$$
4=m+\frac{1}{m}
$$
$$
\begin{array}{ll}
\Rightarrow & m^2-4 m+1 \\
\therefore & m=\frac{4 \pm \sqrt{16-4}}{2}=\frac{4 \pm \sqrt{12}}{2} \\
& m=2 \pm \sqrt{3} \\
\therefore & m_1=2+\sqrt{3} \text { and } m_2=2-\sqrt{3}
\end{array}
$$
Hence, angle between two slopes is
$$
\begin{aligned}
\tan \theta & =\left|\frac{m_1-m_2}{1+m_1 m_2}\right|=\left|\frac{2 \sqrt{3}}{1+1}\right| \\
\Rightarrow \quad \tan \theta & =\sqrt{3}
\end{aligned}
$$
$\begin{aligned} \Rightarrow & \tan \theta & =\tan \frac{\pi}{3} \\ \Rightarrow & \theta & =\frac{\pi}{3}\end{aligned}$
Hence, the correct answer is option: 'B'.