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If P Q R is an equilateral triangle insc...

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

A

center of ellipse

B

focus of ellipse

C

between focus and center on major axis

D

none of these

Text Solution

Verified by Experts

Consider `P(theta),Q(theta+(2pi)/(3)), and R(theta+(4pi)/(3))`. Then ,
`P-=(a cos theta, b sin theta)`
`Q'-=(a cos (theta+(2pi)/(3)), b sin (theta+(2pi)/(3)))`
`and R'-=(a cos (theta+(4pi)/(3)), b sin (theta+(4pi)/(3)))`
Let the centroid of `DeltaP'Q'R`be (x'y')
`x'=a[(cos theta+cos (theta+(2pi)/(3))+cos(theta+(4pi)/(3)))/(3)]`
`y'=(a)/(3)[sin theta+sin (theta+(2pi)/(3))+sin (theta+(4pi)/(3))]=0`
`y'=(a)/(3)[sin theta+sin (theta+(2pi)/(3))+sin (theta+(4pi)/(3))]`
`y'=(a)/(3)[sin theta+2sin(theta+pi)cos.(pi)/(3)]=0`
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