Question

Two tangents are drawn from the point $(-2,-1)$ to the parabola $y^2=4x$. If $\theta$ is the angle between these tangents then $\tan \theta$ equals to

Answer 1

Tangent to the parabola $y^2=4x$,
$y=mx+\frac{1}{m}$
It passes through (-2, -1). Therefore,
$-1=-2m+\frac{1}{m}$
$\Rightarrow 2m^2-m-1=0$
$\Rightarrow (2m+1)(m-1)=0$
$\Rightarrow m=\frac{1}{2},1$
Then the angle between the lines is
$\tan \theta =\left|\frac{m_1+m_2}{1-m_1{m}_2}\right|=3$