Locus of centroid of the triangle whose vertices are (a t,a t), (b t,-b t) and (1,0), wher…

Question

Locus of centroid of the triangle whose vertices are $(a\cos t,a\sin t), (b\sin t,-b\cos t)$ and $(1,0)$, where $t$ is a parameter, is:

Accepted Answer

Centroid $(x,y)$:
$$x=\frac{a\cos t+b\sin t+1}3 \implies a\cos t+b\sin t=3x-1$$
$$y=\frac{a\sin t-b\cos t}3 \implies a\sin t-b\cos t=3y$$

Squaring and adding: $(3x-1)^2+(3y)^2 = (a\cos t+b\sin t)^2+(a\sin t-b\cos t)^2 = a^2+b^2$

Solutions · 1