$\begin{aligned} & x^2-6 x+a=0 (i) \\ & x^2-c x+6=0 (ii)\end{aligned}$
$$
\begin{aligned}
& \frac{\alpha^2}{-36+a c}=\frac{\alpha}{a-6}=\frac{1}{-c+6} \\
& \alpha^2=\frac{a c-36}{6-c}, \alpha=\frac{a-6}{6-c} \\
& \therefore \quad \frac{a c-36}{6-c}=\frac{(a-6)^2}{(6-c)^2} \\
& \Rightarrow \quad(a-6)^2=(6-c)(a c-36)(iii)
\end{aligned}
$$

Also if other roots are in ratio $4: 3$, let other roots are $4 p$ and $3 p$.
$$
\begin{aligned}
& \therefore \quad \alpha \cdot 4 p=a, \alpha \cdot 3 p=6 \\
& \therefore \quad \frac{4}{3}=\frac{a}{6} \Rightarrow a=8
\end{aligned}
$$

Put $a=8$ in eq. (iii)
$$
\begin{aligned}
& (8-6)^2=(6-c)(8 c-36) \\
\Rightarrow & 4=(6-c) \cdot 4(2 c-9) \Rightarrow 12 c-54-2 c^2+9 c=1 \\
\Rightarrow & 2 c^2-21 c+55=0 \Rightarrow 2 c^2-11 c-10 c+55=0 \\
\Rightarrow & c(2 c-11)-5(2 c-11)=0 \Rightarrow(2 c-11)(c-5)=0 \\
\Rightarrow & c=5, \frac{11}{2} \\
\therefore & a=8, c=5
\end{aligned}
$$

Common root $=\frac{a-6}{6-c}=\frac{8-6}{6-5}=2$.