Question

The locus of the point of intersection of two tangents to the parabola $y^2=4ax$, which are at right angle to one another is

• A. $x^2+y^2=a^2$
• B. $a{y}^2=x$
• C. $x+a=0$
• D. $x+y\pm a=0$

Answer 1

Answer: (C)

Let the two tangents to the parabola $y^2=4ax$ be $PT$ and $QT$
which are at right angle to one another at $T(h,k)$.
Then we have a find the locus of $T(h,k)$.
We know that $y=mx+\frac{a}{m}$,
where $m$ is the slope is the equation of tangent to the parabola
$y^2=4ax$ for all $m.$
Since this tangent to the parabola will pass through $T(h,k)$,
so $k=mh+\frac{a}{m};$
or $m^{2}h-mk+a=0$
This is a quadratic equation in $m$,
so will have two roots, say $m_1$ and $m_2$,
then $m_1+m_2=\frac{k}{h}$,
and $m_1\cdot m_2=\frac{a}{h}$
Given that the two tangents intersect at right angle so
$m_1\cdot m_2=-1$
or $\frac{a}{h}=-1$
or $h+a=0$
The locus of $T(h,k)$ is $x+a=0$,
which is the equation of directrix.