Question

Locus of the point of intersection of any two perpendicular tangents to the parabola $x^2=4ay$ is

• A. $x=0$
• B. $y=0$
• C. $x+a=0$
• D. $y+a=0$

Answer 1

The correct option is D $y+a=0$

Equation of tangent with slope $1/m$ to the parabola $x^2=4ay$ is given by,
$x=my+\frac{a}{m}$
$\Rightarrow mx=m^{2}y+a$
$\Rightarrow m^{2}y-mx+a=0$
Above equation is quadratic in $m$
So for tangent to be perpendicular,
Product of roots of the above equation should be $-1$
$\Rightarrow \frac{a}{y}=-1$
$\Rightarrow y=-a$, which is clearly the directrix of the given parabola.