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A variable plane is at a constant distan...

A variable plane is at a constant distance `p` from the origin and meets the coordinate axes in `A , B , C` . Show that the locus of the centroid of the tehrahedron `O A B Ci sx^(-2)+y^(-2)+z^(-2)=16p^(-2)dot`

Text Solution

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Let a variable plane cuts the coordinate axes in A(a, 0, 0), B(0, b, ) and C (0, 0, c) and
`(alpha, beta, gamma)` be the coordinates of centroid of the terahedron OABC.
`therefore alpha = a/4, beta =b/4, gamma =c/4`
` rArr a =4 alpha, b= 4 beta, c=4gamma`
Since the plane `x/a+ y/b + z/c = 1` is at a constant
distance p from origin, hence
`p=(0+0+0-1)/sqrt(1/a^(2)+1/b^(2)+1/c^(2))`
`rArr 1/p^(2)=1/a^(2) +1/b^(2)+1/c^(2)`
`rArr 1/p^(2)=1/((4alpha)^(2))+1/(4beta)^(2)+1/(4gamma)^(2)=1/(16alpha^(2))+1/(16beta^(2))+1/(16gamma^(2))`
Locus of `(alpha, beta, gamma)` is
`16/p^(2)=1/x^(2)+1/y^(2)+1/z^2`
`rArr x^(-2)+y^(-2)+z^(-2)=16p^(_2)`
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