Question

A line $L$ is common tangent to the circle $x ^2+ y ^2=1$ and the parabola $y ^2=4 x$. If $\theta$ is the angle which it makes with the positive $x$-axis, then $\tan ^2 \theta$ is equal to

• A. $2 \sin 18^{\circ}$
• B. $2 \sin 15^{\circ}$
• C. $\cos 36^{\circ}$
• D. $2 \cos 36^{\circ}$

Answer 1

Answer: (A)

Tangent to the parabola
$y^2=4 x$ is $y=m x+\frac{1}{m}$
$m ^2 x - my +1=0,$
As, it touches the circle $x^2+y^2=1$, so
$\left|\frac{1}{\sqrt{ m ^4+ m ^2}}\right|=1 \Rightarrow m ^4+ m ^2-1=0$
$\therefore m ^2=\tan ^2 \theta=\frac{-1 \pm \sqrt{1+4}}{2}=\frac{\sqrt{5}-1}{2}=2\left(\frac{\sqrt{5}-1}{4}\right)=2 \sin 18^{\circ}$