The number of lines that are parallel to $2x+6y+1$ and  have an intercept of length $10$ between co-ordinate axes is  

$2x+6y+\lambda =0$  is parallel to given line 

If a, b be intercepts, then $a=-\frac{\lambda }{2},b=\frac{-\lambda }{6}$ , then 

$a^2+b^2={10}^2$  given 

$\begin{array}{l}\text{ or }\frac{{\lambda }^2}{4}+\frac{{\lambda }^2}{36}=100\Rightarrow 40{\lambda }^2=36\times 4\times 100\\ \therefore \lambda =\pm 6\sqrt{10}\end{array}$

Hence only two lines.