The centroid of the triangle whose three sides are given by the combined equation $$\left( {{x^2} + 7xy + 2{y^2}} \right)\left( {y - 1} \right) = 0$$       is :

Solution :
The sides are $$y=1$$  and the pair $${x^2} + 7xy + 2{y^2} = 0.$$     Clearly, one vertex is $$\left( {0,\,0} \right)$$  and the y-coordinate of each of the other two vertices is 1. Putting $$y=1$$  in the second equation, we get $${x^2} + 7xy + 2 = 0.$$
If $${x_1},\,{x_2}$$  are the roots then $${x_1} + {x_2} = - 7.$$
$$\therefore $$ the centroid $$ = \left( {\frac{{0 + {x_1} + {x_2}}}{3},\,\frac{{0 + 1 + 1}}{3}} \right) = \left( { - \frac{7}{3},\,\frac{2}{3}} \right).$$