The coordinates of a point which divide the line joining the points `P(2,3,1) and Q(5,0,4)` in the ratio `1:2` are

Options

A. `(7//3, 1, 5//3)`
B. `(4,1,3)`
C. `(3,2,2)`
D. `(1,-1,1)`

Answer

To find the coordinates of the point that divides the line segment joining the points \( P(2, 3, 1) \) and \( Q(5, 0, 4) \) in the ratio \( 1:2 \), we can use the section formula in three-dimensional geometry.

### Step-by-Step Solution:

1. **Identify the Points and the Ratio**:
- Let \( P(2, 3, 1) \) be point A with coordinates \( (x_1, y_1, z_1) = (2, 3, 1) \).
- Let \( Q(5, 0, 4) \) be point B with coordinates \( (x_2, y_2, z_2) = (5, 0, 4) \).
- The ratio in which the line segment is divided is \( m:n = 1:2 \).

2. **Apply the Section Formula**:
The coordinates \( (x, y, z) \) of the point dividing the line segment in the ratio \( m:n \) are given by:
\[ x = \frac{n x_1 + m x_2}{m + n}, \quad y = \frac{n y_1 + m y_2}{m + n}, \quad z = \frac{n z_1 + m z_2}{m + n} \]

3. **Calculate the x-coordinate**:
\[ x = \frac{2 \cdot 2 + 1 \cdot 5}{1 + 2} = \frac{4 + 5}{3} = \frac{9}{3} = 3 \]

4. **Calculate the y-coordinate**:
\[ y = \frac{2 \cdot 3 + 1 \cdot 0}{1 + 2} = \frac{6 + 0}{3} = \frac{6}{3} = 2 \]

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

6. **Final Coordinates**:
The coordinates of the point that divides the line segment \( PQ \) in the ratio \( 1:2 \) are \( (3, 2, 2) \).

### Conclusion:
The coordinates of the point are \( (3, 2, 2) \).