Question

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`

Answer 1

Let the equation of the variable plane is
`x/a+y/b+z/c=1`………………… (i)
This plane meets the x-axis, y-axis and z-axis at the points A(a,0,0), B(0,b,0) and C(0,0,c) respectively.
Let `(alpha,beta,gamma)` be the coordinates of the centroid of the tetrahedron OABC.
Then, `alpha(0+a+0+0)/4 =a/4, beta=(0+0+b+0)/4 = b/4` and
`gamma=(0+0+0+c)/4=c/4 rArr a=4alpha, b=4beta, c=4gamma`...............(ii)
`therefore` p=distance of the plane (i) from (0,0,0).
`rArr p=|0/a+0/b+0/c-1|/sqrt(1/a^(2)+1/b^(2)+1/c^(2))`
`rArr 1/p=sqrt(1/a^(2)+1/b^(2)+1/c^(2))=1/p^(2)=(1/a^(2)+1/b^(2)+1/c^(2))`
`rArr 1/p^(2)=1/(16alpha^(2))+1/(16beta^(2))+/1(16gamma^(2))` [using (ii)]
`rArr 1/alpha^(2)+1/beta^(2)+1/gamma^(2)=16/p^(2) rArr alpha^(-2)+beta^(-2)+gamma^(-2)=16p^(-2)`.
Hence, the required locus is `x^(-2)+y^(-2)+z^(-2)=16p^(-2)`.