A variable plane moves so that the sum of reciprocals of its intercepts on the three coordinate axes is constant λ. It passes through a fixed point, which has coordinates
(1λ,1λ.1λ)
Let equation of the variable plane be xa+yb+zc=1 (1)
The intercepts on the coordinate axes are a, b , c. The sum of reeciprocals of intercepts is constant λ, therefore
1a+1b+1c=λ or (1λ)a+(1λ)b+(1λ)c=1∴(1λ,1λ,1λ)
lies on the plane (1).
Hence, the variable plane (1) always passes
through the fixed point (1λ,1λ.1λ).