Question

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

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

Answer 1

Answer: (A)

Tangent to the parabola
$y^2=4x$ is $y=mx+\frac{1}{m}$
$m^{2}x−my+1=0,$
As, it touches the circle $x^2+y^2=1$, so
$â\frac{1}{\sqrt{m^4+m^2}}â=1⇒m^4+m^2−1=0$
$∴m^2={\tan }^2θ=\frac{−1Â\pm \sqrt{1+4}}{2}=\frac{\sqrt{5}−1}{2}=2\left(\frac{\sqrt{5}−1}{4}\right)=2\sin 18^{∘}$