Question

$A(5,2),B(-3,5)$ and vertex C is the point on the locus $y=2+3x+x^2$, Find the equation of the locus of centroid G of $△ABC$.

Answer 1

Let the centroid be h,b

$h=\frac{5-3+x}{3}$

$b=\frac{2+5+y}{3}$

Centroid is given by $h=(\frac{(x_1+x_2+x_3)}{3},\frac{(y_1+y_2+y_3)}{3})$

$h=\frac{2+x}{2}$

$x=3h-2$

$y=3b-7$

Put x and y in $y=2+3x+x^2$

$3b-7=2+3(3h-2)+(3h-2{)}^2$

Replace $b=y$, $h=x$

$3y-7=2+3(3x-2)+(3x-2{)}^2$ is the required locus.