The coordinates of the point which divides the line segment joining the points (5,4,2) and (-1,-2,4) in the ratio of 2:3 externally is

Options

A. `(17,16,-2)`
B. none of these
C. `((13)/(5),(8)/(5),(14)/(5))`
D. `((17)/(5),(16)/(5),(2)/(5))`

Answer

To find the coordinates of the point that divides the line segment joining the points \( A(5, 4, 2) \) and \( B(-1, -2, 4) \) in the ratio of \( 2:3 \) externally, we can use the formula for external division.

### Step-by-Step Solution:

1. **Identify the Coordinates and Ratios**:
- Let \( A(x_1, y_1, z_1) = (5, 4, 2) \)
- Let \( B(x_2, y_2, z_2) = (-1, -2, 4) \)
- The ratio \( m:n = 2:3 \) (where \( m = 2 \) and \( n = 3 \))

2. **Use the External Division Formula**:
The formula for the coordinates \( P(x, y, z) \) that divides the line segment externally in the ratio \( m:n \) is given by:
\[ x = \frac{mx_2 - nx_1}{m - n}, \quad y = \frac{my_2 - ny_1}{m - n}, \quad z = \frac{mz_2 - nz_1}{m - n} \]

3. **Calculate the x-coordinate**:
\[ x = \frac{2 \cdot (-1) - 3 \cdot 5}{2 - 3} = \frac{-2 - 15}{-1} = \frac{-17}{-1} = 17 \]

4. **Calculate the y-coordinate**:
\[ y = \frac{2 \cdot (-2) - 3 \cdot 4}{2 - 3} = \frac{-4 - 12}{-1} = \frac{-16}{-1} = 16 \]

5. **Calculate the z-coordinate**:
\[ z = \frac{2 \cdot 4 - 3 \cdot 2}{2 - 3} = \frac{8 - 6}{-1} = \frac{2}{-1} = -2 \]

6. **Combine the Coordinates**:
The coordinates of the point \( P \) that divides the line segment externally in the ratio \( 2:3 \) are:
\[ P(17, 16, -2) \]

### Final Answer:
The coordinates of the point are \( (17, 16, -2) \).

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