If the gradient of one of the lines ${\mathrm{x}}^2+\mathrm{hxy}+2{\mathrm{y}}^2=0$ is twice that of the other, then sum of possible values of h is_____

Let ${\mathrm{m}}_1$and ${\mathrm{m}}_2$ be the slopes ${\mathrm{x}}^2+\mathrm{hxy}+2{\mathrm{y}}^2=0.$ Then, ${\mathrm{m}}_1+{\mathrm{m}}_2=-\frac{\mathrm{h}}{2},{\mathrm{m}}_1{\mathrm{m}}_2=\frac{1}{2}$

But ${\mathrm{m}}_1=2{\mathrm{m}}_2$  (given). Therefore, $3{\mathrm{m}}_2=-\frac{\mathrm{h}}{2}$ and $2{\mathrm{m}}_2^2=\frac{1}{2}$

i,e., ${\mathrm{m}}_2^2=\frac{1}{4}$ Also ${\mathrm{m}}_2=-\frac{\mathrm{h}}{6}$ 

$\therefore \frac{{\mathrm{h}}^2}{36}=\frac{1}{4}$ or ${\mathrm{h}}^2=9$ or $\mathrm{h}=\pm 3$