Explanation:

- The mean proportional (geometric mean) is found by:

$$x=\sqrt{12.8\times 64.8}$$

- The third proportional (in a:b=c:d format, with b = c) is found by:

$$y=\frac{(57.6{)}^2}{38.4}$$

- Calculate $x$:

- $x=\sqrt{12.8\times 64.8}$

- $x=\sqrt{829.44}$

- $x=28.8$

- Calculate $y$:

- $y=\frac{57.6\times 57.6}{38.4}$

- $y=\frac{3317.76}{38.4}$

- $y=86.4$

- Find $2x:y$:

- $2x=2\times 28.8=57.6$

- Ratio $2x:y=57.6:86.4$ simplifies to $2:3$

- Correct Option:

- $$ \boxed{\textcolor{green}{\text{ 2 : 3}}} $$