Question

If $A(\cos X,\sin X),B(\sin X,-\cos X),C(-2,1)$ are the vertices of a $△ABC,$ then as $A$ varies the locus of its centroid is:

• A. $x^2+y^2-2x-4y+1=0$
• B. $3(x^2+y^2)-2x-4y+1=0$
• C. $x^2+y^2-2x-4y+3=0$
• D. none of these

Answer 1

The correct option is D none of these

Given vertices

$A(\cos X,\sin X),B(\sin X,-\cos X),C(-2,1)$

Let Centroid $G(h,k)$

$h=\frac{\cos X+\sin X-2}{3}$

$k=\frac{\sin X-\cos X+1}{3}$

$h+k=\frac{\cos X+\sin X-2+\sin X-\cos X+1}{3}$

$h+k=\frac{\cos X+\sin X-2+\sin X-\cos X+1}{3}$

$h+k=\frac{2\sin X-1}{3}$

Putting $h=x,k=y$ in above equation

$x+y-\frac{2\sin X-1}{3}=0$

$3x+3y-2\sin X+1=0$