AB, AC are tangents to a parabola `y^(2)=4ax`. If `l_(1),l_(2),l_(3)` are the lengths of perpendiculars from A, B, C on any tangent to the parabola, then

Options

A. `l_(1), l_(2), l_(3)` are in GP
B. `l_(2), l_(1), l_(3)` are in GP
C. `l_(3), l_(1), l_(2)` are in GP
D. `l_(3), l_(2), l_(1)` are in GP

Answer

B, C

Explanation

Let the coordinates of B C be `(at_(1)^(2), 2at_(1))` and `(at_(1)t_(2), a(t_(1)+t_(2))` respectively. Then, the coordinates of A are `(at_(1)t_(2), a(t_(1)+t_(2))`
The equation of any tangent to `y^(2)=4ax` is `ty=x+aty^(2)`
`:." "l_(1)=(at_(1)t_(2)-a(t_(1)-t_(2))t+at^(2))/(sqrt(1+t^(2)))`
Clearly, `l_(2)l_(3)=l_(1)^(2)`. Therefore, `l_(2)l_(1)l_(3)` are om G.P.