A line L has intercepts a and b on the coordinate axes. When the axes are rotated through an angle, keeping the origin fixed, the same line L has intercepts p and q. Then,

Since the line L has intercepts a and b on the coordinate
axes, therefore its equation is
$\frac{x}{a}+\frac{y}{b}=1$                                                (1)
When the axes are rotated, its equation with respect to
the new axes and same origin will become
$\frac{x}{p}+\frac{y}{q}=1$                                             (2)
In both the cases, the length of the perpendicular from the origin to the line will be same.
$\therefore \frac{1}{\sqrt{{\displaystyle \frac{1}{a^2}+\frac{1}{b^2}}}}=\frac{{\displaystyle 1}}{{\displaystyle \sqrt{\frac{1}{{{\displaystyle p}}^2}+\frac{1}{q^2}}}}$

$or \frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{p^2}+\frac{1}{q^2}$
which is the required relation.