Browse Questions

# 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,prove that$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{1}{{a'}^2}+\frac{1}{{b'}^2}+\frac{1}{{c'}^2}$

Toolbox:
• Perpendicular diatance of the plane $ax+by+cz+d=0$ from origin is $\bigg|\large\frac{d}{\sqrt {a^2+b^2+c^2}}\bigg|$
• If two system of lines have the same origin then their $\perp$ distance from origin to the plane in both the system are equal.
Let the equation of the plane in both the systems be
$\large\frac{x}{a}+\large\frac{y}{b}+\large\frac{z}{c}$$=1 and \large\frac{X}{a'}+\large\frac{Y}{b'}+\large\frac{Z}{c'}$$=1$
We know that $\perp$ diatance of the plane $ax+by+cz+d=0$ from origin is $\bigg|\large\frac{d}{\sqrt {a^2+b^2+c^2}}\bigg|$
It is given that the origin is same for both the system.
Hence $\perp$ distance of the plane from origin are equal for both the system.