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Q)

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

$\begin{array}{1 1}(A)\;\frac{\pi}{2} \\(B)\;\frac{\pi}{3} \\(C)\;\frac{\pi}{4} \\(D)\;\frac{\pi}{6} \end{array}$

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A)
Any tangent to $y^2=4x$ is $y=mx +\large\frac{1}{m}$ It passes through $(1,4)$
$4=m+\large\frac{1}{m}$
=> $m^2-4m-1=0$
=> $m_1+m_2 =4 m_1m_2 =1$
$\therefore |m_1-m_2|= 2 \sqrt 3$
$\tan \theta =\large\frac{2 \sqrt 3}{1+1}$
$\qquad=\sqrt 3$
$\theta= \large\frac{\pi}{3}$
Hence B is the correct anwer.
M^2−4M−1=0 is wrong
M^2-4M+1=0 is correct
Remaining answer is correct
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