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Question

A variable plane remains at constant distance p from the origin. If it meets coordinates axes at points A,B,C then the locus of the centroid of ABC is

A
x2+y2+z2=9p2
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B
x3+y3+z3=9p3
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C
x2+y2+z2=9p2
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D
x3+y3+z3=9p3
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Solution

The correct option is A x2+y2+z2=9p2

Let the variable plane be xa+yb+zc=1

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

Let G(α,β,γ) be the centroid of ABC

α=a3,β=b3,γ=c3 .........(1)

Also given that, distance of plane from origin is p

11a2+1b2+1c2=p

1a2+1b2+1c2=1p2

1α2+1β2+1γ2=9p2 using (1)

Hence, required locus of G(α,β,γ) is,

1x2+1y2+1z2=9p2

i.e. x2+y2+z2=9p2


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