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 coordinateaxes, therefore its equation isxa+yb=1                                                (1)When the axes are rotated, its equation with respect tothe new axes and same origin will becomexp+yq=1                                             (2)In both the cases, the length of the perpendicular from the origin to the line will be same.∴11a2+1b2=11p2+1q2or   1a2+1b2=1p2+1q2which is the required relation.