Two perpendicular striaght lines through the focus of the parabola y²=4ax meets its directrix in T₁ and T₂ respectively. Show that the tangent the tangent to the parabola parallel to the perpendicular lines intersect in the midpoint of T₁T₂.

Take the slopes of 2 perpendicular SL as m and -1/m

Now write eqn of both lines as it passes through focus (a,0).

Now we can get T1 and T2 both points as they are intersection of Lines with directix x = -a

Once you getT1 and T2 then you can find its midpoint also.

Finally you need to write eqn of tangents and find its intersection point.

I have given you all the steps , calculation you can do

Thanks