Question

Given the parabola $y^2=4ax,$ find the locus of intersection of pair of tangents that are perpendicular to each other

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let (h,k) be the locus of the point

We know that pair of tangents from(h,k) to the parabola $y^2=4ax$ will be $T^2=S{S}^{\prime }$

$T=yk-2a(x+h)$

$S=y^2-4ax$

$S^{\prime }=k^2-4ah$

$(yk-2a(x+h){)}^2=(y^2-4ax\left)\right(k^2-4ah)$

In this equation of pair of straight lines to be perpendicular sum of coefficients of $x^2$ and $y^2$ should be 0.

Therefore, $4ah+4a^2=0$

$\Rightarrow h=-a$

Therefore the locus of intersection of pair of tangents those are perpendicular to each other

is $x=-a.$ This is also called the director circle in general for conic sections and in case of parabola its same as directrix.