Two systems of rectangular axes have the same origin. If a plane cuts them at distances `a ,b ,ca n da^(prime),b^(prime),c^(prime)` respectively, prove that `1/(a^2)+1/(b^2)+1/(c^2)=1/(a^('2))+1/(b^('2))+1/(c^('2))`

Explanation

Let `a,b,c` be the intercepts when `Ox,Oy,Oz` are taken as axes;
then the equation of the plane is
`x/a+y/b+z/c=1`
Also, let `a^(prime),b^(prime),c^(prime) `be the intercepts when `OX,OY,OZ` are taken as axes; then in this case equation of the same plane is ,
`X/a^(prime)+Y/B^(prime)+Z/C^(prime)=1`
Now these are equations of the same plane and in both the cases the origin is same.

Hence, length of the perpendicular drawn form the origin to the plane in both the case must be the same,
that is ,
`1/sqrt(1/(a^2)+1/(b^2)+1/(c^2))=1/sqrt(1/(a^('2))+1/(b^('2))+1/(c^('2)))`
`=>1/(a^2)+1/(b^2)+1/(c^2)=1/(a^('2))+1/(b^('2))+1/(c^('2))`