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 1a2−1p2+1b2−1q2 is equal to

Equation of the line L in the two coordinate systems is xa+yb=1,Xp+Yq=1where  (X, Y) are the new coordinates of a point (x, y) 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.11a2+1b2=11p2+1q2⇒1a2+1b2=1p2+1q2or 1a2−1p2+1b2−1q2=0