Question

A variable plane moves in such a way that the sum of reciprocals of its intercepts on the three coordinate axes is constant. Show that the plane passes through a fixed point.

Answer 1

Eq. of Plane

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

Intercepts of plane

$A(a,0,0),B(0,b,0),C(0,0,c)$

$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$

$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$

So $x=\frac{1}{k},y=\frac{1}{k}=z=\frac{1}{k}$