Constant Term Condition

General term in the expansion is

$$T_{r+1}=\binom{12}{r}x^{12-r}\left(\frac3{x^2}\right)^r$$

$$=\binom{12}{r}3^r x^{12-3r}$$

Coefficient of

$$x^9$$

occurs when

$$12-3r=9$$

$$3r=3$$

$$r=1$$

Hence corresponding term is

$$T_2=\binom{12}{1}3^1x^9$$

Therefore coefficient of

$$x^9$$

is

$$12\times3=36$$

Now term independent of

$$x$$

occurs when

$$12-3r=0$$

$$r=4$$

Hence constant term is

$$T_5=\binom{12}{4}3^4$$

$$=495\times81$$

Therefore required ratio is

$$36:495\times81$$

Divide by

$$9$$

$$=4:4455$$

Hence, the required ratio is

$$\boxed{4:4455}$$