If $H(\alpha ,\beta ),G({\alpha }_1,{\beta }_1)$ and $C({\alpha }_2,{\beta }_2)$ be the orthocenter, centroid and circumcentre respectively of the triangle formed by the lines connected by $2xy=x^{2}y+x{y}^2.$ Then the value of $5\alpha -4\beta +3{\alpha }_1+3{\beta }_1-2{\alpha }_2-{\beta }_2$ is___

$2xy=x^{2}y+x{y}^2$

$2xy-x^{2}y-x{y}^2=0$

$xy(2-x-y)=0$

$x=0,y=0,x+y=2$ are the sides of triangle.

Triangle is right angled 

Centroid $x=0,y=0,x+y=2$

Hence, circum centre. 

$C({\alpha }_2,{\beta }_2)=(\frac{(2+\frac{0}{2})}{2},\frac{0+2}{2})=(1,1)$

Orthocenter $H(\alpha ,\beta )=(0,0)$

$5\alpha -4\beta +3{\alpha }_1+3{\beta }_1-2{\alpha }_2-{\beta }_2=5\times 0-4\times 0+3\times \frac{2}{3}+3\times \frac{2}{3}-2\times 1-1=2+2-2-1=1$