A sphere of constant radius `2k` passes through the origin and meets the axes in `A ,B ,a n dCdot` The locus of a centroid of the tetrahedron `O A B C` is a. `x^2+y^2+z^2=4k^2` b. `x^2+y^2+z^2=k^2` c. `2(k^2+y^2+z)^2=k^2` d. none of these

Options

A. `x^(2)+y^(2)+z^(2)=k^(2)`
B. `x^(2)+y^(2)+z^(2)=k^(2)`
C. `2(k^(2)+y^(2)+z)^(2)=k^(2)`
D. none of these

Answer

b

Explanation

Let the equation of the sphere be `x^(2)+y^(2)+z^(2)-ax-by-cz=0`. This meets the axes at `A(a, 0, 0), B(0, b, 0) and C(0, 0, c)`.
Let `(alpha, beta, gamma)` be the coordinatres of the centroid of the tetrahedron OABC. Then
`" "(a)/(4)=alpha, (b)/(4)=beta, (c)/(4) = gamma`
or `" "a= 4alpha, b = 4beta, c= 4gamma`
Now, radius of the sphere = `2k`
`rArr" "(1)/(2)sqrt(a^(2)+b^(2)+c^(2))=2k or a^(2)+ b^(2)+c^(2)= 16k^(2)`
or `" "16(alpha^(2) + beta^(2) + gamma^(2))= 16k^(2)`
Hence, the locus of `(alpha, beta, gamma)` is `(x^(2)+y^(2)+z^(2))=k^(2)`