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

Consider `OX,OY,OZ ` and `"ox",yy,oz` are two system of rectangular axes.
Let their corresponding equation of plane be
`x/a+y/b+z/c=1` …………..i
and `x/(a')+y/(b')+z/(c')=1` …………….ii
Also, the length of perpendicular from origin of Eqs (i) and (ii) must be same.
`:. (0/a+0/b+0/c-1)/(sqrt(1/(a^(2))+1/(b^(2))+1/(c^(2))))=(0/(a')+0/(b')+0/(c')-1)/(sqrt(1/(a^(2))+1/(b^(2))+1/(c^(2)))`
`implies sqrt(1/(a^('2))+1/(b^('2))+1/(c^('2)))=sqrt(1/(a^(2))+1/(b^(2))+1/(c^(2)))`
`implies 1/(a^(2))+1/(b^(2))+1/(c^(2))=1/(a^('2))+1/(b^('2))+1/(c^('2))`