The points $A\left(1 ,3\right) and C\left(5 ,1\right)$ are opposite vertices of rectangle ABCD . If the slope of BD is 2 then its equation 

Given points $A\left(1 ,3\right) and C\left(5 ,1\right)$

$$\begin{gathered} mid point of AC=\left(\frac{1+5}{2} , \frac{3+1}{2}\right)=\left(3 , 2\right) \\ and given slope of BD=2 \end{gathered}$$

In rectangle diagonals are bisect each other

$$\begin{gathered} Let \left(3 ,2\right)=\left(x_1 ,y_1\right) and m=2 \\ equatin of BD is y-y_1=m\left(x-x_1\right) \\ \Rightarrow y-2=2\left(x-3\right) \\ \Rightarrow 2x-y=4 \end{gathered}$$