Let N be the foot of perpendicular drawn from the focus S of any parabola to the tangent at any point P. If O be the vertex of the parabola, then

OS, SN, SP are in A.P.

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OS, SN, SP are in GP.

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OS, SN, SP are in H.P.

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None of these

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Solution

The correct option is A OS, SN, SP are in GP.
Eqn. of tangent at P, $t y = x +, a t^{2}$
$\Rightarrow N \equiv (0, - a t)$
$S N^{2} = a^{2} + a^{2} t^{2} = a^{2} (1 + t)^{2} = O S . S P$
$\Rightarrow$ OS, SN, SP are in G.P.

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Q is a point on the auxiliary circle of an ellipse. P is the corresponding point on ellipse. N is the foot of perpendicular from focus S, to the tangent of auxillary circle at Q. Then

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Q. If T be any point on the tangent at any point P of a parabola, and if TL be perpendicular to the focal radius SP and TN be perpendicular to the directrix, prove that SL = TN.

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Let N be the foot of perpendicular drawn from the focus S of any parabola to the tangent at any point P. If O be the vertex of the parabola, then

The correct option is A OS, SN, SP are in GP.Eqn. of tangent at P, ty=x+,at2⇒N≡(0,−at)SN2=a2+a2t2=a2(1+t)2=OS.SP⇒ OS, SN, SP are in G.P.

The correct option is A OS, SN, SP are in GP.
Eqn. of tangent at P, $t y = x +, a t^{2}$
$\Rightarrow N \equiv (0, - a t)$
$S N^{2} = a^{2} + a^{2} t^{2} = a^{2} (1 + t)^{2} = O S . S P$
$\Rightarrow$ OS, SN, SP are in G.P.