If `P Q R` is an equilateral triangle inscribed in the auxiliary circle of the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1,(a > b),` and `P^(prime)Q^(prime)R '` is the correspoinding triangle inscribed within the ellipse, then the centroid of triangle `P^(prime)Q^(prime)R '` lies at center of ellipse focus of ellipse between focus and center on major axis none of these

Explanation

As, `PQR` is an equilateral triangle inscribed in the auxiliary circle of the ellipse, its vertices will be,

`P(theta),Q(theta+2pi/3) and R(theta+4pi/3)`

Similarly, `P'Q'R'` is the correspoinding triangle inscribed within the ellipse,its vertices wil be,

`P'(acostheta,bsintheta),Q'(acos(theta+2pi/3),bsin(theta+2pi/3)) and R'(acos(theta+4pi/3),bsin(theta+4pi/3))`

So, centroid C(x,y) can be calculated as

`x = (acostheta+acos(theta+2pi/3)+acos(theta+4pi/3))/3`

`x = a/3(costheta+cos(theta+2pi/3)+cos(theta+4pi/3))`

By applying the formula of `cosC+cosD`,

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