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

The correct option is C $x+a=0$

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 $m_1.m_2=\frac{a}{h}$
Given that the two tangents intersect at right angle so
$m_1.m_2=-1$ 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.