Question

From a variable point on the tangent at the vertex of a parabola `y^2=4a x ,` a perpendicular is drawn to its chord of contact. Show that these variable perpendicular lines pass through a fixed point on the axis of the parabola.

Answer 1

Equation of chord of contact to parabola `y^(2)=4ax` w.r.t.
point `P(0,lamda)` is
`lamday=2ax`
`or2ax-lamday=0` (1)
Equation of line perpendicular to this line is `lamdax+2ay=c`.
If is passes through `p(0,lamda)` then `c=2alamda`.
Therefore, equation of line perpendicular to (1) through point P is
`lamdax+2ay=2alamda`
`or2ay+lamda(x-2a)=0`
This equation represents the equation of family of straight lines concurrent at point of intersection of lines y=0andx-2a=0.
So, lines are concurrent at (2a,0).