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`

Explanation

To solve the problem, we need to find the locus of the centroid of the tetrahedron formed by the origin \( O \) and the points \( A, B, C \) where the variable plane meets the coordinate axes. The plane is at a constant distance \( p \) from the origin.

### Step 1: Equation of the Plane
The equation of a plane at a distance \( p \) from the origin can be expressed as:
\[ \frac{Ax + By + Cz + D}{\sqrt{A^2 + B^2 + C^2}} = p \]
For simplicity, we can take \( D = -p \) and rewrite the equation as:
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