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Question

If β is one of the angles between the normals to the ellipse, x2+3y2=9 at the points (3cosθ,3sinθ) and (3sinθ,3cosθ); θ(0,π2); then 2cotβsin2θ is equal to

A
23
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B
13
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C
2
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D
34
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Solution

The correct option is A 23
x2+3y2=9
equation of the normal at (3cosθ,3sinθ)
and (3sinθ,3cosθ) will be
3xcosθ3ysinθ=6
and 3xsinθ3ycosθ=6
Let m1,m2 be their slopes, then

m1=3tanθ,m2=3cotθ
So,
tanβ=m1m21+m1m2tanβ=3tanθ+3cotθ13tanθ3cotθtanβ=32|sinθcosθ|tanβ=3sin2θ1cotβ=3sin2θcotβsin2θ=13
2cotβsin2θ=23

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