If the slope of one line of the pair of lines represented by ${\mathrm{ax}}^2+10\mathrm{xy}+{\mathrm{y}}^2=0$ is four times the slope of the other, then a is equal to 

If ${\mathrm{m}}_1,{\mathrm{m}}_2$ are the slopes of two lines whose combined equation is ${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2=0$ then the relation between the slopes and the coefficients of equation are  ${\mathrm{m}}_1+{\mathrm{m}}_2=-\frac{2\mathrm{h}}{\mathrm{b}},{\mathrm{m}}_1{\mathrm{m}}_2=\frac{\mathrm{a}}{\mathrm{b}}$
Since the slopes of the lines represented by the equation ${\mathrm{ax}}^2+10\mathrm{xy}+{\mathrm{y}}^2=0$ are one is four times of the other. 
Hence,$\mathrm{m}+4\mathrm{m}=10\Rightarrow \mathrm{m}=-2$ and $\mathrm{m}\cdot 4\mathrm{m}=\mathrm{a}\Rightarrow \mathrm{a}=16$
Therefore, $\mathrm{a}=\boxed{16}$