A variable plane forms a tetrahedron of constant volume `64k^3` with the coordinate planes and the origin, then locus of the centroid of the tetrahedron is

Options

A. `x^3+y^3+z^3=6k^3`
B. `xyz=6k^3`
C. `x^2+y^2+z^2=4k^2`
D. `x^(-2)+y^(-2)+z^(-2)=4k^(-2)`

Answer

To solve the problem of finding the locus of the centroid of a tetrahedron formed by a variable plane with a constant volume of \(64k^3\) with the coordinate planes and the origin, we can follow these steps:

### Step 1: Understand the Volume of the Tetrahedron
The volume \(V\) of a tetrahedron formed by the coordinate planes and a point \((x, y, z)\) is given by the formula:
\[ V = \frac{1}{6} \times x \times y \times z \]
where \(x\), \(y\), and \(z\) are the intercepts of the plane with the coordinate axes.

### Step 2: Set Up the Volume Equation
Given that the volume is constant and equal to \(64k^3\), we can set up the equation:
\[ \frac{1}{6} \times x \times y \times z = 64k^3 \]
Multiplying both sides by 6 gives:
\[ x \times y \times z = 384k^3 \]

### Step 3: Find the Centroid of the Tetrahedron
The centroid \(G\) of a tetrahedron with vertices at the origin \(O(0, 0, 0)\) and points \(A(x, 0, 0)\), \(B(0, y, 0)\), and \(C(0, 0, z)\) is given by:
\[ G\left(\frac{x}{4}, \frac{y}{4}, \frac{z}{4}\right) \]

### Step 4: Express the Coordinates of the Centroid
Let the coordinates of the centroid be \(G(X, Y, Z)\):
\[ X = \frac{x}{4}, \quad Y = \frac{y}{4}, \quad Z = \frac{z}{4} \]
This implies:
\[ x = 4X, \quad y = 4Y, \quad z = 4Z \]

### Step 5: Substitute into the Volume Equation
Substituting \(x\), \(y\), and \(z\) in terms of \(X\), \(Y\), and \(Z\) into the volume equation:
\[ (4X)(4Y)(4Z) = 384k^3 \]
This simplifies to:
\[ 64XYZ = 384k^3 \]

### Step 6: Simplify the Equation
Dividing both sides by 64 gives:
\[ XYZ = 6k^3 \]

### Conclusion: Locus of the Centroid
The locus of the centroid \(G(X, Y, Z)\) is given by the equation:
\[ XYZ = 6k^3 \]
This represents the locus of the centroid of the tetrahedron as the plane varies.