Question

The locus of the point of intersection of perpendicular tangents to the parabola $y^2=4ax$ is

• A. Latus rectum
• B. Directrix
• C. Tangent at the vertex
• D. Axis of the parabola

Answer 1

The correct option is B Directrix

Equation of tangent with slope $m$ to the parabola $y^2=4ax$ is given by,

$y=mx+\frac{a}{m}$

$\Rightarrow my=m^{2}x+a$

$\Rightarrow m^{2}x-ym+a=0$

Above equation is quadratic in $m$

So for tangent to be perpendicular,

Product of roots of the above equation should be $-1$

$\Rightarrow \frac{a}{x}=-1$

$\Rightarrow x=-a$, which is clearly the directrix of the given parabola