Question

A variable plane is at a constant distance 'r' from the origin and meets the axes in A, B, C then, the locus of the centroid of the tetrahedron OABC is–

• A. + + = $\frac { 16 } { \mathrm { r } ^ { 2 } }$
• B. x² + y² + z² = 4r
• C. x² + y² + z² = 16r²
• D. x² + y² + z² = 9r²

Answer 1

Answer: (A)

+ + = $\frac { 16 } { \mathrm { r } ^ { 2 } }$

Answer 2

Let the plane be $\frac { \mathrm { x } } { \mathrm { a } }$ + $\frac { \mathrm { y } } { \mathrm { b } }$ + = 1

perpendicular distance from origin to variable plane = r

so r =$\left| \frac { 1 } { \sqrt { \frac { 1 } { \mathrm { a } ^ { 2 } } + \frac { 1 } { \mathrm {~b} ^ { 2 } } + \frac { 1 } { \mathrm { c } ^ { 2 } } } } \right|$ Ž =

Let G ŗ (x₁, y₁, z₁) be centroid of tetrahedron OABC

So x₁ = a/4 Ž a = 4x₁ similarly b = 4y₁, c = 4z₁

So + + =

Ž + + =