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

Explanation

Let the equation of the variable plane be
`x/a+y/b+z/c=1`…………….(i)
Then, it makes intercepts a,b,c with the coordinate axes.
`therefore 1/a+1/b+1/c=k,` where k is a constant (given)
`rArr 1/(ka)+1/(kb)+1/(kc)=1 rArr 1/a(1/k)+1/b(1/k)+1/c(1/k)=1`.
`rArr (1/k,1/k,1/k)` satisfies (i).
Hence, the given plane passes through a fixed point `(1/k,1/k,1/k)`.