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

Let the roots of equation $x^2-6x+a=0$  be   $\alpha$ and  $4\beta$   and that of the equation

$x^2-cx+6=0$ be  $\alpha$ and   $4\beta$. Then 

$\alpha +4\beta =6;4\alpha \beta =a$

and $\alpha +3\beta =c;3\alpha \beta =6$

$\Rightarrow a=8$

The equation becomes  $x^2-6x+8=0$

$\Rightarrow (x-2)(x-4)=0$

~ roots are 2 and 4 

 $\Rightarrow \alpha =2,\beta =1\therefore \text{ Common root is }2$