The number of lines that are parallel to $2x+6y+7=0$ and have an intercept of length 10 between the coordinate axes, is

The equation of any line parallel to $2x+6y+7=0$ is

$2x+6y+k=0$. This meets the axes at $A(-k/2,0)$ and 

$B(0,-k/6)$.

Now,

$\begin{array}{l} AB=10\\ \Rightarrow \sqrt{\frac{k^2}{4}+\frac{k^2}{36}}=10\\ \Rightarrow \sqrt{\frac{10k^2}{36}}=10\\ \Rightarrow 10k^2=3600\Rightarrow k=\pm 6\sqrt{10}\end{array}$

Hence, there are two lines given by $2x+6y\pm 6\sqrt{10}=0$