Let P be an arbitrary point on the ellipse `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1, a gt b gt 0`. Suppose `F_(1)` and `F_(2)` are the foci are the ellipse. The locus of the centroid of the triangle `PF_(1)F_(2)` as P moves on the ellipse is- (A) a circle (B) a parabola (C) an ellipse (D) a hyperbola

Options

A. a circle
B. a parabola
C. an ellipse
D. a hyperbola

Answer

C

Explanation

`P rarr a cos theta, b sin theta`
` G rarr ((sumx_(i))/(3),(sumy_(i))/(3))`
`F_(i) rarr(ae,0) , F_(2) rarr (-ae,0)`
`h = (acostheta + ae - ae)/(3) , cos theta = (3h)/(a)`
`k = (b sin theta)/(3), sin theta = (3k)/(b)`
`cos^(2)theta + sin^(2)theta = 1`
`(9h^(2))/(a^(2)) + (9k^(2))/(b^(2)) = 1 rArr (x^(2))/((a^(2)//9)) + (y^(2))/((b^(2)//9)) = 1`
(Ellipse)