Question

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

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

Answer 1

Answer: (A)

Given,$y^2=4x[\therefore a=1]$
General equation of parabola $=y=mx+\frac{a}{m}$
[where, $m$ is the slope $]$
at $(1,4);4=m+\frac{a}{m}$
$\Rightarrow 4=m+\frac{1}{m}\Rightarrow m^2-4m+1=0$
Let $m_1$ and $m_2$ are the roots.
$\therefore m_1+m_2=4$ and $m_1{m}_2=1$
Now, $\tan \theta =\frac{|m_1-m_2|}{|1+m_1{m}_2|}=\frac{\sqrt{{\left(m_1+m_2\right)}^2-4m_1{m}_2}}{1+m_1{m}_2}$
$=\frac{\sqrt{(4{)}^2-4(1)}}{1+1}=\frac{\sqrt{16-4}}{2}=\frac{\sqrt{12}}{2}=\frac{2\sqrt{3}}{2}$
$\Rightarrow \tan \theta =\sqrt{3}\Rightarrow \tan \theta =\tan \frac{\pi }{3}\Rightarrow \theta =\frac{\pi }{3}$