Question

Which of the following represents the
director circle for the parabola $y^2=4ax?$

• A. $x^2+y^2=a^2$
• B. $x^2+y^2=1$
• C. $x+a=0$
• D. $x=a$

Answer 1

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

The locus of point of intersection of perpendicular tangents from a parabola is the director circle.

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 should be 0.

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

$\therefore h=-a$

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

is $x=-a.$

In case of parabola this is nothing but the directrix which can be viewed as a circle with infinite radius. For the parabola $y^2=4ax$ this is
$x+a=0.$