If the G.M of two non-zero positive numbers is to their A.M is $12:13$  then numbers are in the ratio

Let the numbers be $a,b$

$\frac{\text{G}\text{.M}}{\text{A}\text{.M}}=\frac{12}{13}\Rightarrow \frac{\sqrt{ab}}{\left(\frac{a+b}{2}\right)}=\frac{12}{13}$

$\frac{2\sqrt{ab}}{a+b}=\frac{12}{13}$

$\frac{a+b}{2\sqrt{ab}}=\frac{13}{12}$

$\frac{{\left(a+b\right)}^2}{4ab}=\frac{169}{144}\cdots \cdots \cdots \left(1\right)$

$\frac{{\left(a+b\right)}^2}{4ab}-1=\frac{169}{144}-1$

$\frac{{\left(a-b\right)}^2}{4ab}=\frac{25}{144}\cdots \cdots \cdots \left(2\right)$

$\frac{\frac{{\left(a+b\right)}^2}{4ab}}{\frac{{\left(a-b\right)}^2}{4ab}}=\frac{\frac{169}{144}}{\frac{25}{144}}$

$\frac{a+b}{a-b}=\frac{13}{5}$

By componendo and dividendo

$\frac{\left(a+b\right)+\left(a-b\right)}{\left(a+b\right)-\left(a-b\right)}=\frac{13+5}{13-5}$

$\frac{2a}{2b}=\frac{18}{8}\Rightarrow \frac{a}{b}=\frac{9}{4}$

$\therefore$ The numbers are in the ratio $9:4\left(\text{or}\right)4:9$