The centroid of the triangle formed by the lines  $x+y=1$  and  $2y^2-xy-6x^2=0$ is

 $2y^2-xy-6x^2=0\Rightarrow (y-2x)(2y+3x)=0$
$\therefore$  Sides of triangle are  $x+y=1,y-2x=0,2y+3x=0$
$\therefore$  Vertices are  $(\frac{1}{3},\frac{2}{3}),(0,0),(-2,3)$
$\therefore$  Centroid is  $(\frac{-5}{9},\frac{11}{9})$