O,A,B are vertices of triangle whose sides are given by  ${\left(5\mathrm{x}+12\mathrm{y}\right)}^2-3{\left(12\mathrm{x}-5\mathrm{y}\right)}^2=0$  $5x+12y-78=0$ then  AB=

The combined equation of pair of lines which forms an equilateral triangle with the line $5x+12y-78=0$  is  $(5\mathrm{x}+12\mathrm{y}{)}^2-3(12\mathrm{x}-5\mathrm{y}{)}^2=0$
The area of the equilateral triangle is $\frac{{\mathrm{c}}^2}{\sqrt{3}\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)}=\frac{(78{)}^2}{\sqrt{3}\left(5^2+{12}^2\right)}=\frac{36}{\sqrt{3}}\text{ square units }$ 
If  be the length of the side of the equilateral triangle then the area of the triangle is  $\frac{\sqrt{3}}{4}{a}^2$
It implies that 
$\begin{array}{c}\frac{\sqrt{3}}{4}{\mathrm{a}}^2=\frac{36}{\sqrt{3}}\\ {\mathrm{a}}^2=\frac{144}{3}\\ \mathrm{a}=\frac{12}{\sqrt{3}}\\ =6.9282\end{array}$
Therefore, the length of the side is $\boxed{6.9282}$