Question

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

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{3}$
• C. $\frac{2\pi }{5}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (B)

Equation of a tangent to a parabola $y^2=4x$ is
$y=mx+\frac{1}{m}$
Since, it passes through point $(1,4)$. So,
$4=m+\frac{1}{m}$
or $4m=m^2+1$
$\Rightarrow m^2-4m+1$
$\therefore m=\frac{4\pm \sqrt{16-4}}{2}=\frac{4\pm \sqrt{12}}{2}$
$m=2\pm \sqrt{3}$
$\therefore m_1=2+\sqrt{3}$ and $m_2=2-\sqrt{3}$
Hence, angle between two slopes is
$\tan \theta =\left|\frac{m_1-m_2}{1+m_1{m}_2}\right|$
$=\left|\frac{2\sqrt{3}}{1+1}\right|$
$\Rightarrow \tan \theta =\sqrt{3}$
$\Rightarrow \tan \theta =\tan \frac{\pi }{3}$
$\Rightarrow \theta =\frac{\pi }{3}$