Find the co-ordinates of a point which divides the line segment joining P(5, 4, 2) and Q(-1, -2, 4) in the ratio 2 : 3.

Explanation

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

### Step-by-Step Solution:

1. **Identify the coordinates of points P and Q**:
- Let \( P = (x_1, y_1, z_1) = (5, 4, 2) \)
- Let \( Q = (x_2, y_2, z_2) = (-1, -2, 4) \)

2. **Identify the ratio in which the line segment is divided**:
- The ratio is given as \( m:n = 2:3 \), where \( m = 2 \) and \( n = 3 \).

3. **Apply the section formula**:
The coordinates of the point R that divides the line segment joining P and Q in the ratio \( m:n \) are given by:
\[ R\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \]

4. **Substituting the values into the formula**:
- For the x-coordinate:
\[ x = \frac{2 \cdot (-1) + 3 \cdot 5}{2 + 3} = \frac{-2 + 15}{5} = \frac{13}{5} \]

- For the y-coordinate:
\[ y = \frac{2 \cdot (-2) + 3 \cdot 4}{2 + 3} = \frac{-4 + 12}{5} = \frac{8}{5} \]

- For the z-coordinate:
\[ z = \frac{2 \cdot 4 + 3 \cdot 2}{2 + 3} = \frac{8 + 6}{5} = \frac{14}{5} \]

5. **Combine the coordinates**:
Therefore, the coordinates of the point R that divides the line segment in the ratio 2:3 are:
\[ R\left( \frac{13}{5}, \frac{8}{5}, \frac{14}{5} \right) \]

### Final Answer:
The coordinates of the point R are \( \left( \frac{13}{5}, \frac{8}{5}, \frac{14}{5} \right) \).

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