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Global Maxima

If the tangen...

Question

If the tangents at $P$ and $Q$ on a parabola meet in $T,$ then $S P, S T$ and $S Q$ are in

A . P .

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G . P .

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H . P .

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None of the above

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Solution

The correct option is B $G . P .$
If $P$ and $Q$ are $t_{1}$ and $t_{2},$ then $T$ is $[{a t}_{1} t_{2}, a (t_{1} + t_{2})]$
$S P = a + x = a (1 + t_{1}^{2}), S Q = a (1 + t_{2}^{2}) S T^{2} = a^{2} {(1 + t_{1} t_{2})}^{2} + a^{2} {(t_{1} + t_{2})}^{2}$
${S T}^{2} = a^{2} (1 + t_{1}^{2} + t_{2}^{2} + t_{1}^{2} t_{2}^{2}) {S T}^{2} = a (1 + t_{1}^{2}) . a (1 + t_{2}^{2}) = S P . S Q$

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If the tangents at P and Q in the parabola meet in T, then which of the following statements are correct.

1. TP and TQ subtends equal angle at the focus S

2. $S T^{2}$ = SP.SQ

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