Question

A variable plane is at a constant distance p from the origin and meets the axes in A, B and C. The locus of the centroid of the tetrahedron $OABC$ is

• A. $x^{- 2} + y^{- 2} + z^{- 2} = 16p^{- 2}$
• B. $x^{- 2} + y^{- 2} + z^{- 2} = 16p^{- 1}$
• C. $x^{- 2} + y^{- 2} + z^{- 2} = 16$
• D. None of these

Answer 1

Answer: (A)

$x^{- 2} + y^{- 2} + z^{- 2} = 16p^{- 2}$

Answer 2

Plane is $\frac { x } { a } + \frac { y } { b } + \frac { z } { c } = 1$ , where $p = \frac { 1 } { \sqrt { \sum \left( \frac { 1 } { a ^ { 2 } } \right) } }$

or $\frac { 1 } { a ^ { 2 } } + \frac { 1 } { b ^ { 2 } } + \frac { 1 } { c ^ { 2 } } = \frac { 1 } { p ^ { 2 } }$ …..(i)

Now according to equation,

Put the values of x, y, z in (i), we get the locus of the centroid of the tetrahedron.