The quadratic equations x² – 6x + a = 0 and x² – 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 which of the following is(are) correct?

Let 

$$\begin{gathered} \mathrm{Now}, \mathrm{\alpha }+4\mathrm{\beta }=6 \text{ and }4\mathrm{\alpha \beta }=\mathrm{a} \\ \mathrm{\alpha }+3\mathrm{\beta }=\mathrm{c}\text{ and }3\mathrm{\alpha \beta }=6 \\ \text{ We get }\mathrm{\alpha \beta }=2 \end{gathered}$$
a = 8.
So, the first equation is
x² – 6x + 8 = 0
x = 2, 4
$$\begin{gathered} \text{If }\mathrm{\alpha }=2\text{ and }4\mathrm{\beta }=4\text{, then }\mathrm{\beta }=1. \\ \text{If }\mathrm{\alpha }=4\text{ and }4\mathrm{\beta }=2,\text{ then }\mathrm{\beta }=\frac{1}{2} \end{gathered}$$
$\Rightarrow 3\mathrm{\beta }=\frac{3}{2}$, which is non-integer.
The common root is 2