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If a variable plane forms a tetrahedron ...

If a variable plane forms a tetrahedron of constant volume `64k^3` with the co-ordinate planes, then the locus of the centroid of the tetrahedron is:

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The correct Answer is:
`xyz=6k^(3)`
Let the variable plane intersect the coordinate axes at `A(a,0,b,),B(0,b,0)andC(0,0,c)`. Then the equation of the plane will be
`(x)/(a)+(y)/(b)+(z)/(c)=1" "(i)`
Let `P(alpha,beta,gamma)` be the centroid of tetrahedron OABC. Then,
`alpha=(a)/(4),beta=(b)/(4)andgamma=(c)/(4)`
or `a=4alpha,b=4betaandc=4gamma`
`implies` Volume of tetrahedron `("Area of "DeltaAOB)OC`
or `64k^(3)=(1)/(3)((1)/(2)ab)c=(abc)/(6)`
`=((4alpha)(4beta)(4gamma))/(6)ork^(3)=(alphabetagamma)/(6)`
Therefore, the required locus of `P(alpha,beta,gamma)` is `xyz=6^(3)`
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