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Question

If the tangent at the point P(θ) to the ellipse 16x2+11y2=256 is also a tangent to the circle x2+y22x=15, then possible value(s) of θ is/are

A
2π3
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B
4π3
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C
5π3
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D
π3
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Solution

The correct options are
C 5π3
D π3
Given circle is x2+y22x=15
x22x+1+y2=15+1(x1)2+y2=42

Ellipse is 16x2+11y2=256
For ellipse, equation of tangent at (4cosθ,1611sinθ) is:
16x(4cosθ)+11y(1611sinθ)=256x(4cosθ)+11y(111sinθ)=16

It touches the circle (x1)2+y2=42
So, distance of tangent from centre of circle is 4
∣ ∣4cosθ1616cos2θ+11sin2θ∣ ∣=4(cosθ4)2=16cos2θ+11sin2θ4cos2θ+8cosθ5=0cosθ=12 or cosθ=52(not possible)
θ=2nπ±π3,nZ

Hence, possible values of θ are π3,5π3

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