Question

A variable plane remains at constant distance $p$ from the origin.If it meets coordinate axes at points $A , B , C$ then the locus of thecentroid of $\Delta ABC$ is

• A. $x^{-2}+y^{-2}+z^{-2}=9 p^{-2}$
• B. $x^{-3}+y^{-3}+z^{-3}=9 p^{-3}$
• C. $x^{2}+y^{2}+z^{2}=9 p^{2}$
• D. $x^{3}+y^{3}+z^{3}=9 p^{3}$

Answer 1

Answer: (A)

Let $A \equiv( a , 0,0), B \equiv(0, b , 0), C \equiv(0,0, c )$,
then equation of the plane is
$\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$
Its distance from the origin,
$\frac{1}{ a ^{2}}+\frac{1}{ b ^{2}}+\frac{1}{ c ^{2}}=\frac{1}{ p ^{2}} \ldots$ (i)
If $(x, y, z)$ be centroid of $\Delta A B C$, then
$x =\frac{ a }{3}, y =\frac{ b }{3}, z =\frac{ c }{3} \dots$(ii)
Eliminating $a , b , c$ from (i) and (ii) required locus is
$x^{-2}+y^{-2}+z^{-2}=9 p^{-2}$