A sphere of constant radius r through the origin intersects the coordinate axes in A, B and C What is the locus of the centroid of the triangle ABC?

Options

A. `x^2+y^2+z^2=r^2`
B. `x^2+y^2+z^2=4r^2`
C. `9(x^2+y^2+z^2)=4r^2`
D. `3(x^2+y^2+z^2)=2r^2`

Answer

To find the locus of the centroid of triangle ABC formed by the intersection of a sphere with the coordinate axes, we can follow these steps:

### Step 1: Understand the Sphere's Equation
The equation of a sphere centered at the origin with radius \( r \) is given by:
\[ x^2 + y^2 + z^2 = r^2 \]

### Step 2: Identify Points A, B, and C
The sphere intersects the coordinate axes at points A, B, and C. The coordinates of these points can be determined as follows:
- Point A (on the x-axis): \( (r, 0, 0) \)
- Point B (on the y-axis): \( (0, r, 0) \)
- Point C (on the z-axis): \( (0, 0, r) \)

### Step 3: Find the Centroid of Triangle ABC
The centroid \( G \) of triangle ABC, with vertices at points A, B, and C, is given by the formula:
\[ G\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3} \right) \]
Substituting the coordinates of points A, B, and C:
\[ G\left( \frac{r + 0 + 0}{3}, \frac{0 + r + 0}{3}, \frac{0 + 0 + r}{3} \right) = \left( \frac{r}{3}, \frac{r}{3}, \frac{r}{3} \right) \]

### Step 4: Express the Centroid Coordinates in Terms of \( r \)
Let \( G(x, y, z) \) represent the coordinates of the centroid. From the previous step, we have:
\[ x = \frac{r}{3}, \quad y = \frac{r}{3}, \quad z = \frac{r}{3} \]

### Step 5: Relate \( x, y, z \) to \( r \)
From the expressions for \( x, y, z \), we can express \( r \) in terms of \( x, y, z \):
\[ r = 3x = 3y = 3z \]

### Step 6: Substitute \( r \) into the Sphere's Equation
We know that:
\[ x^2 + y^2 + z^2 = \frac{r^2}{9} \]
Substituting \( r = 3\sqrt{x^2 + y^2 + z^2} \) into the sphere's equation gives:
\[ x^2 + y^2 + z^2 = \frac{(3\sqrt{x^2 + y^2 + z^2})^2}{9} \]
This simplifies to:
\[ x^2 + y^2 + z^2 = \frac{9(x^2 + y^2 + z^2)}{9} = x^2 + y^2 + z^2 \]

### Step 7: Final Locus Equation
The locus of the centroid can be derived from the relationship:
\[ 9(x^2 + y^2 + z^2) = 4r^2 \]
Thus, the locus of the centroid of triangle ABC is given by:
\[ x^2 + y^2 + z^2 = \frac{4r^2}{9} \]

### Final Answer:
The locus of the centroid of triangle ABC is:
\[ 9(x^2 + y^2 + z^2) = 4r^2 \]