Question

The locus of a point such that two tangents drawn from it to the parabola $y^2=4ax$ are such that the slope of one is double the other is -

• A. $y^2=\frac{9}{2}ax$
• B. $y^2=\frac{9}{4}ax$
• C. $y^2=9ax$
• D. $x^2=4ay$

Answer 1

Answer: (A)

Let the point be $(h,k)$
Now equation of tangent to the parabola $y^2=4ax$ whose slope is $m$ is
$y=mx+\frac{a}{m}$
as it passes through $(h,k)$
$\therefore k=mh+\frac{a}{m}\Rightarrow m^{2}h-mk+a=0$
It has two roots $m_1,2m_1$
$\therefore m_1+2m_1=\frac{k}{h},2m_1\cdot m_1=\frac{a}{h}$
$m_1=\frac{k}{3h}\dots \text{ (i) }$
$m_1^2=\frac{a}{2h}\dots \text{ (ii) }\text{ from (i) \& (ii) }$
$\Rightarrow \frac{k^2}{(3h{)}^2}=\frac{a}{2h}\Rightarrow k^2=\frac{9a}{2}h$
Thus locus of point is $y^2=\frac{9}{2}ax$.