Let the two tangents to the parabola
y2=4ax be PT andQT which are at right angle to one another at T(h, k).
Then we have to find the locus of T(h, k).
We know that
y=mx+,where m is the slope is the equation of tangent to the parabola
y2=4ax for all m
Since this tangent to the parabola will pass through T(h, k) so
This is a quadratic equation in m so will have two roots,say
m1 and
m2, then
m1+m2=,andm1:m2=Given that the two tangents intersect at right angle so
The locus of T(h, k) is
x+a=0, which is the equation of directrix.