First slide

Parabola

Question

The length of the perpendiculars from the focus
and the extremities of a focal chord of a parabola
on the tangent at the vertex form

Moderate

A

an A.P.
B

a G.P.
C

an H.P.
D

none of these.

Solution

$y^{2} = 4 a x$ be the parabola, $P (a t^{2}, 2 a t), Q (\frac{a}{t^{2}}, \frac{- 2 a}{t})$
and the focus is $S (a, 0)$ The tangent at the vertex
is $x = 0 .$ Lengths of the perpendicular from $P$ ,
$Q$ and $S$ on this tangent are respectively $l_{1} = a t^{2}$ ,
$l_{2} = a l t^{2}, l_{3} = a$ so that $l_{1} l_{2} = a^{2} = l_{3}^{2} \Rightarrow l_{1}, l_{2},$
$l_{3}$ form a G.P.

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