The quadratic equations ${\mathrm{x}}^2-6\mathrm{x}+\mathrm{a}=0$ and ${\mathrm{x}}^2-\mathrm{cx}+6=0$ have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. Then the common root is 

Let the roots of ${\mathrm{x}}^2-6\mathrm{x}+\mathrm{a}=0\text{ be }\mathrm{\alpha }, 4\mathrm{\beta }$ and that of ${\mathrm{x}}^2-\mathrm{cx}+6=0\text{ be }\mathrm{\alpha }, 3\mathrm{\beta }$

$\therefore 4\mathrm{\alpha \beta }=\mathrm{a}\text{ and }3\mathrm{\alpha \beta }=6\Rightarrow \mathrm{a}=8$

$\therefore$ First equation becomes

${\mathrm{x}}^2-6\mathrm{x}+8=0 \Rightarrow \mathrm{x}=2, 4$

$\Rightarrow 4\mathrm{\beta }$ is either 2 or 4, but $4\mathrm{\beta }\ne 2$ (since, other roots are given integers) 

$\therefore \mathrm{\alpha }=2$

(i.e.) common root is 2