Question

Equation Of the locus of the centroid of the triangle

Answer 1

Calculate the centroid of a triangle with three vertices by using formula: C = [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3].

Substitute (x1, y1) = (a cos t, a sin t), (x2, y2) = (b sin t, -b cos t) and (x3, y3) = (1, 0) in the above formula.

C = [(a cos t + b sin t + 1)/3, (a sin t - b cos t + 0)/3]

C = [(a cos t + b sin t + 1)/3, (a sin t - b cos t)/3]

Answer 2

locus of the centroid of a triangle whose vertices are (a cost, a sint),
 (b sint, -b cost)
 and (1,0), where t is the parameter 
Let (h, k) be the centroid of the triangle 
h=a cos t + b sin t + 13 and k=a sin t −b cos t +03 a cos t + b sin t = 3h −1 and a sin t − b cos t = 3k ⇒(a cos t + b sin t)2+( a sin t − b cos t)2=(3h−1)2+9k2 ⇒a2+b2=(3h−1)2+9k2⇒9x2+9y2−6x−a2−b2+1=0