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Focal Chord

If PSP' is ...

Question

If $P S P^{'}$ is a focal chord of the parabola $y^{2} = 4 a x,$ and $S L$ is its semi latus rectum then, $S P, S L, S P^{'}$ are in

A.P

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G.P

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H.P

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A.G.P

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Solution

The correct option is C H.P
Let $P = (a t^{2}, 2 a t), ∴ P^{'} = (\frac{a}{t^{2}}, - \frac{2 a}{t})$
$S = (a, 0), L = (a, 2 a)$
Now $S P = \sqrt{(a - a t^{2})^{2} + (2 a t)^{2}} = a + a t^{2}$
$S P^{'} = \sqrt{{(a - \frac{a}{t^{2}})}^{2} + {(\frac{- 2 a}{t})}^{2}} = a + \frac{a}{t^{2}} = \frac{a + a t^{2}}{t^{2}}$
Now clearly,
$\frac{1}{S P} + \frac{1}{S P^{'}} = \frac{2}{S L}$
Hence, $S P, S L, S P^{'}$ are in H.P

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