Question

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

• A. $\frac{\pi }{3}$
• B. ${\tan }^{-}1\left(12/\sqrt{5}\right)$
• C. ${\tan }^{-}1\left(5\sqrt{5}/7\right)$
• D. none of these

Answer 1

The correct option is A $\frac{\pi }{3}$

We know that , any tangent to $y^2=4x$ is $y=mx+1/m$. As it passes through $(1,4)$

So, satisfying

$4=m+1/m$

$m^2-4m-1=0$

Let the roots be $m_1$ and $m_2$

$m_1+m_2=4$

$m_1+m_2=1$

$(m1-m2{)}^2=(m1+m2{)}^2-4m1m2$

$|m1-m2|=2\sqrt{3}$

$\tan \theta =\left|\frac{m1-m2}{1+m1m2}\right|$

$\tan \theta =3\sqrt{3}/2$

$\tan \theta =\sqrt{3}$

$\theta =\pi /3$