Thank you for reporting, we will resolve it shortly
Q.
The locus of a point such that two tangents drawn from it to the parabola $y^2=4 a x$ are such that the slope of one is double the other is -
Conic Sections
Solution:
Let the point be $( h , k )$
Now equation of tangent to the parabola $y^2=4 a x$ whose slope is $m$ is
$y=m x+\frac{a}{m}$
as it passes through $( h , k )$
$\therefore k = mh +\frac{ a }{ m } \Rightarrow m ^2 h - mk + a =0$
It has two roots $m_1, 2 m_1$
$\therefore m _1+2 m _1=\frac{ k }{ h }, 2 m _1 \cdot m _1=\frac{ a }{ h } $
$ m _1=\frac{ k }{3 h } \ldots \text { (i) } $
$ m _1^2=\frac{ a }{2 h } \ldots \text { (ii) } \text { from (i) \& (ii) }$
$\Rightarrow \frac{ k ^2}{(3 h )^2}=\frac{ a }{2 h } \Rightarrow k ^2=\frac{9 a }{2} h$
Thus locus of point is $y ^2=\frac{9}{2} ax$.