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Q.

P is a point on the ellipse x2a2+y2b2=1 with foci at S,Sl . Normal at P cuts the x-axis at G of  SPSlP=23 then SGSlG=

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a

2a3b

b

23

c

49

d

32

answer is D.

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Detailed Solution

Question Image

Given ellipse x2a2+y2b2=1(a>b)

Let P=a cosθ, b sinθ be any point on ellipse.

Equation of normal at P(θ) is axcosθ-bysinθ=a2-b2

since normal cuts x-axis at G then G=(a2-b2)cosθa, 0

Now CG=(a2-b2)cosθa

and SP=a(1-e cosθ), S1P=a(1+e cosθ)

Now SPS1P=a(1-e cosθ)a(1+e cosθ)=23  1-e cosθ1+e cosθ=23

Now SG=CS-CG

               =ae-(a2-b2)cosθa=a2e-a2e2cos θa

SG=a2e(1-e cosθ)a=ae(1-ecosθ)

similarly  S1G=ae(1+e cosθ)

Now SGS1G=1-e cosθ1+e cosθ=23

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P is a point on the ellipse x2a2+y2b2=1 with foci at S,Sl . Normal at P cuts the x-axis at G of  SPSlP=23 then SGSlG=