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Question

A variable tangent to the circle x2+y2=1 intersects the ellipse 2x2+y2=4 at P and Q. The locus of the point of intersection of the tangents at P and Q is

A
an ellipse of eccentricity 34
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B
a parabola with focus at (2,3)
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C
an ellipse of area 8π sq. units
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D
a parabola with latus rectrum 12
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Solution

The correct option is D an ellipse of area 8π sq. units
R.E.F image
Let y=mx+c is tangent to x2+y2=1
Cm2+1=1
C=m2+1
y=mx+m2+1 ...(1)
Let P(2cosα,2sinα),Q(2cosβ,2sinβ)
Since, p lies on (1)
2sinα=m×2cosα+4cos2α+1
PQ as a
T=0
y×h4+x×k2=1
kx2hy4120
mxy+m2+1=0
k2m=h4=1m2+1
h=4m2+1,k=2mm2+1
(h4)2+(k2)2=1m2+1+m2m2+1
h216+k24=1
x216+y24=1
It represent ellipse A=abπ=4×2×π
A=8π


1092253_1190516_ans_eb768980eab848e89181591107036b8f.png

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