The line  $L$ has intercepts  $a$ and $b$  on the coordinate axes. 

The coordinate axes are rotated through a fixed angle, keeping the origin fixed. 

If $p$  and $q$  are the intercepts of the line $L$  on the new axes,

then $\frac{1}{a^2}-\frac{1}{p^2}+\frac{1}{b^2}-\frac{1}{q^2}$ is equal to 

Equation of the line $L$ in the two coordinate system is   $\frac{x}{a}+\frac{y}{b}=1,$$\frac{X}{p}+\frac{Y}{q}=1$
Where $\left(X,Y\right)$ are the new coordinates of a point  $\left(x,y\right)$ when the axes are rotated through a fixed angle, keeping the origin fixed. As the length of the perpendicular from the origin has not changed 
$\frac{1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}}}=\frac{1}{\sqrt{\frac{1}{p^2}+\frac{1}{q^2}}}\Rightarrow \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}+\frac{1}{q^2}$
or  $\frac{1}{a^2}-\frac{1}{p^2}+\frac{1}{b^2}-\frac{1}{q^2}=0.$