Discussion Forum



1)

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 the centroid  of $\triangle$  ABC is  


A) $x^{-2}+y^{-2}+z^{-2}=9p^{-2}$

B) $x^{-3}+y^{-3}+z^{-3}=9p^{-3}$

C) $x^{2}+y^{2}+z^{2}=9p^{2}$

D) $x^{3}+y^{3}+z^{3}=9p^{3}$

Answer:

Option A

Explanation:

Let A = (a,0,0) , B= (0,b,0) C= (0,0,c) then equation of the plane is 

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

 Its distance from the origin,

 $\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{1}{c^{2}}=\frac{1}{p^{2}}$  ......(i)

 If (x,y,z) the centroid of $\triangle$ ABC, then

$x=\frac{a}{3},y=\frac{b}{3},z=\frac{c}{3}$   .....(ii)

eliminating a,b,c from (i) and (ii) required locus is 

 $x^{-2}+y^{-2}+z^{-2}=9p^{-2}$