Two systems of rectangular axis have the same origin. If a plane cuts them at distances a,b,c and a',b;c', respectively from the origin, then prove that 1a2+1b2+1c2=1a′2+1b′2+1c′2.
Consider OX,OY,OZ and ox,oy,oz are two system of rectangluar axes.
Let their corresponding equation of plane be xa+yb+zc=1 ...(i)
and xa′+yb′+zc′=1 ...(ii)
Also, the length of perpendicular from origin to Eqs. (i) and (ii) must be same. ∴0a+0b+0c−1√1a2+1b2+1c2=0a′+0b′+0c′−1√1a2+1b2+1c2⇒=√1a′2+1b′2+1c′2=√1a2+1b2+1c2⇒1a2+1b2+1c2=1a′2+1b′2+1c′2