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 P′Q′R′ is corresponding triangle inscribed within the ellipse. Then centroid of the triangle P′Q′R′ 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 ellipseLet 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 △P′Q′R′≡(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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