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. $ay^2 =x$
• C. $x + a = 0$
• D. $x + y \; \pm \; a = 0$

Answer 1

Answer: (C)

$x + a = 0$

Answer 2

Let the two tangents to the parabola $y^{2}=4 a x$ be $P T$ and $Q T$ 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 $y=m x+\frac{a}{m}$, where $m$ is the slope is the equation of tangent to the parabola $y ^{2}=4 ax$ 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.