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Question

If the normal at any point P on an ellipse of eccentricity e meets the major axis and minor axis at T and G respectively, then PT:PG equal to

A
e
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B
1:e
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C
1e2:1
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D
1e2:e
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Solution

The correct option is D 1e2:1
Ellipse x2a2+y262=1
Equation of normal at point P(acosθ,bsinθ)
is axsecθ6ycscθ=a2b2
9+ cuts on xaxis
x=a2b2asecθ T(a2b2asecθ,0)
9+ cuts yaxis at
y=aa2b2bsinθCr.(0,b2a2bcscθ)
PT= acosθ(a2b2a)2+62sin2θ
PG= a2cos2θ+(bsinθ(b2a2bcscθ))2
PTPG=b4a2cos2θ+62sin2θ
a2cos2θ+a4b2sin2θ
PTPG=   (1e2)b2cos2θ+b2sin2θ(1a21e2)sin2θ+a2cos2θ=1e2b2e2b2cos2θb2e2e2cos2θ
PTPG=1e2b2b2=1e21e2=1e2:1

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