Practicing Success
Two system of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b', c' from the origin, then |
$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{a'^2}-\frac{1}{b'^2}-\frac{1}{c'^2}=0$ $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{a'^2}+\frac{1}{b'^2}+\frac{1}{c'^2}=0$ $\frac{1}{a^2}+\frac{1}{b^2}-\frac{1}{c^2}+\frac{1}{a'^2}+\frac{1}{b'^2}-\frac{1}{c'^2}=0$ $\frac{1}{a^2}-\frac{1}{b^2}-\frac{1}{c^2}-\frac{1}{a'^2}-\frac{1}{b'^2}-\frac{1}{c'^2}=0$ |
$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{a'^2}-\frac{1}{b'^2}-\frac{1}{c'^2}=0$ |
The equation of the plane with reference to the two systems of rectangular axes are $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}= 1$ ..............(i) and, $\frac{X}{a'}+\frac{Y}{b'}+\frac{Z}{c'}= 1$ ............(ii) Since the origin is same. ∴ Length of the ⊥ from (0, 0, 0) on (i) = Length of the ⊥ from (0, 0, 0) on (ii) $⇒ \begin{vmatrix} \frac{1}{\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}}\end{vmatrix}=\begin{vmatrix} \frac{1}{\sqrt{\frac{1}{a'^2}+\frac{1}{b'^2}+\frac{1}{c'^2}}}\end{vmatrix}$ $⇒\frac{1}{a^2}+ \frac{1}{b^2}+\frac{1}{c^2}-\frac{1}{a'^2}-\frac{1}{b'^2}-\frac{1}{c'^2}=0$ |