The coordinates of a point which divides the join of
points (3, 3, 7) and (8, 3, 1) internally in the ratio
`2 : 1` is

Options

A. `(19/3, 3, 3)`
B. `(1, -2, 3)`
C. `(3, 3, 0)`
D. `(19/3, -3, -3)`

Answer

To find the coordinates of the point that divides the line segment joining the points \( P(3, 3, 7) \) and \( Q(8, 3, 1) \) internally in the ratio \( 2:1 \), we can use the section formula in three-dimensional geometry.

### Step-by-Step Solution:

1. **Identify the Points and the Ratio**:
- Let \( P = (3, 3, 7) \) and \( Q = (8, 3, 1) \).
- The ratio in which the point divides the line segment is \( 2:1 \).

2. **Apply the Section Formula**:
The coordinates of the point \( M \) that divides the line segment joining points \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) in the ratio \( m:n \) are given by:
\[ M = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]
Here, \( m = 2 \) and \( n = 1 \).

3. **Calculate the x-coordinate**:
\[ x = \frac{2 \cdot 8 + 1 \cdot 3}{2 + 1} = \frac{16 + 3}{3} = \frac{19}{3} \]

4. **Calculate the y-coordinate**:
\[ y = \frac{2 \cdot 3 + 1 \cdot 3}{2 + 1} = \frac{6 + 3}{3} = \frac{9}{3} = 3 \]

5. **Calculate the z-coordinate**:
\[ z = \frac{2 \cdot 1 + 1 \cdot 7}{2 + 1} = \frac{2 + 7}{3} = \frac{9}{3} = 3 \]

6. **Combine the Coordinates**:
The coordinates of the point \( M \) that divides the segment in the ratio \( 2:1 \) are:
\[ M = \left( \frac{19}{3}, 3, 3 \right) \]

### Final Answer:
The coordinates of the point that divides the line segment joining the points \( (3, 3, 7) \) and \( (8, 3, 1) \) internally in the ratio \( 2:1 \) are \( \left( \frac{19}{3}, 3, 3 \right) \).