The graph $xy=1$ is reflected in $y = 2x$ to give the graph of the curve $12x^2+rxy+s{y}^2+t=0$, then the number of prime factors of $\frac{rst}{20}=\mathrm{........}$

Let the image of $A(\alpha ,\beta )$ about $y=2xbeB(a,b)$ mid point of $\overline{AB}=(\frac{\alpha +a}{2},\frac{\beta +b}{2})$ lie on  y - 2x = 0
$\Rightarrow \beta +b=2\alpha +2a\mathrm{............}\left(1\right)$ and slope of 

$\overline{AB}=\frac{-1}{2}\Rightarrow \frac{\beta -b}{\alpha -b}=\frac{-1}{2}\Rightarrow \beta -b=\frac{-1}{2}a+\frac{a}{2}\mathrm{...........}\left(2\right)$

(1) and (2)  $\Rightarrow 2b=\frac{5}{2}a+\frac{3}{2}\alpha \Rightarrow \alpha =\frac{4b-3a}{5}\Rightarrow B=\frac{8b-6a}{5}+2a-b=\frac{3b+49}{5}$
(2) $(\alpha ,\beta )$  lie on xy = 1  $\Rightarrow \frac{4b-3a}{5}.\frac{3b+4a}{5}=1$
$\begin{array}{l}\Rightarrow 12b^2+7ab-12a^2=25\\ \Rightarrow 12x^2-7xy-12y^2+25=0\\ \Rightarrow r=-7,s=-12,t=25\end{array}$