Prove that the locus of the centroid of the triangle whose vertices are `(acost ,asint),(bsint ,-bcost),` and `(1,0)` , where `t` is a parameter, is circle.

Options

A. `(3x+1)^(2) + (3y)^(2) = a^(2) - b^(2)`
B. `(3x - 1)^(2) = a^(2) - b^(2)`
C. `(3x -1)^(2) + (3y)^(2) = a^(2) + b^(2)`
D. `(3x + 1)^(2) + (3y)^(2) = a^(2) + b^(2)`

Explanation

Let (h , k) be the coordinates of the centroid . Then ,
h = `( a cos t + b sin t + 1)/(3) ` and `k = (a sin t - b cos t + 0)/(3)`
`implies 3h - 1 = a cos t + b sin t ` and 3k = a sin t - b cos t
`implies (3h-1)^(2) + (3k)^(2) = (a cos t + b sint)^(2) + (a sin t- b cos t)^(2)`
`implies (3h-1)^(2) + (3k)^(2) = a^(2) + b^(2)`
Hence , the locus of (h , k) is `(3x-1)^(2) + (3y)^(2) = a^(2) + b^(2)`