The quadratic equations $x^2-6x+a=0$  and $x^2-cx+6=0$  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 $x^2-6x+a=0$  be $\alpha ,4\beta$  and that of $x^2-cx+6=0$  be $\alpha \text{ and }3\beta$  
$\therefore \alpha +4\beta =6\text{ and }4\alpha \beta =a$ and  $\therefore \alpha +3\beta =c\text{ and }3\alpha \beta =6$
 $\Rightarrow \frac{a}{6}=\frac{4}{3}\Rightarrow a=8$  $\therefore x^2-6x+8=0$
 $\Rightarrow (x-4)(x-2)=0\Rightarrow x=2,4$
 And  $x^2-cx+6=0$
 $$\begin{gathered} \Rightarrow 2^2-2c+6=0\Rightarrow c=5 \\ \therefore x^2-5x+6=0 \\ \Rightarrow x=2,3 \end{gathered}$$
 Hence, common root is 2.