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Question

Let P,Q,R be the points on the auxiliary circle of ellipse x2a2+y2b2=1(a>b) , such that PQR is an equilateral triangle and PQR is corresponding triangle inscribed within the ellipse. Then centroid of the triangle PQR lies at

A
centre of the ellipse
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B
focus of the ellipse
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C
between focus and centre of the ellipse
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D
between one extremity of minor axis and centre of the ellipse
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Solution

The correct option is A centre of the ellipse
Let points on equilateral triangle be triangleP(θ),Q(θ+2π3),R(θ+4π3)
then
P(acosθ,bsinθ)Q(acos(θ+2π3),bsin(θ+2π3))R(acos(θ+4π3),bsin(θ+4π3))

Let centroid of PQR(x,y)
x=a⎢ ⎢ ⎢ ⎢cos(θ)+cos(θ+2π3)+cos(θ+4π3)3⎥ ⎥ ⎥ ⎥x=a3[cosθ+2cos(θ+π)cosπ3]=0

And
y=b⎢ ⎢ ⎢ ⎢sin(θ)+sin(θ+2π3)+sin(θ+4π3)3⎥ ⎥ ⎥ ⎥y=b3[sinθ+2sin(θ+π)cosπ3]=0

Hence centroid is (0,0), which is the centre of the ellipse.

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