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))`

Options

A. `1/(a^(2))+1/(b^(2))+1/(c^(2))-1/(a^('2))-1/(b^('2))-1/(c^('2))=0`
B. `1/(a^(2))+1/(b^(2))+1/(c^(2))+1/(a^('2))+1/(b^('2))+1/(c^('2))=0`
C. `1/(a^(2))+1/(b^(2))-1/(c^(2))+1/(a^('2))+1/(b^('2))-1/(c^('2))=0`
D. `1/(a^(2))-1/(b^(2))-1/(c^(2))+1/(a^('2))-1/(b^('2))-1/(c^('2))=0`

Answer

A

Explanation

The equations of the plane with reference to the two systems of rectangular axes are
`x/a+y/b+z/c=1`……………..i and
`X/(a')+Y/(b')+Z/(c')=1`……………..ii
Since the origin is same.
`:.` Length of the `_|_` from (0,0,0) on i
`=` Length of the `_|_` from (0,0,0) on ii
`implies|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))=0`