Find the co-ordinates of the point which divides the join of the points (-1,2,3) and (4,-2,5) in the ratio 1 : 2 externally.

Answer

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

### Step-by-Step Solution:

1. **Identify the Points and the Ratio:**
- Let the points be \( A(-1, 2, 3) \) and \( B(4, -2, 5) \).
- The ratio in which the point divides the line segment externally is \( m_1:m_2 = 1:2 \).

2. **Apply the Section Formula for External Division:**
- The coordinates \( (x, y, z) \) of the point that divides the line segment externally in the ratio \( m_1:m_2 \) can be calculated using the formula:
\[ x = \frac{m_1 x_2 - m_2 x_1}{m_1 - m_2}, \quad y = \frac{m_1 y_2 - m_2 y_1}{m_1 - m_2}, \quad z = \frac{m_1 z_2 - m_2 z_1}{m_1 - m_2} \]
- Here, \( (x_1, y_1, z_1) = (-1, 2, 3) \) and \( (x_2, y_2, z_2) = (4, -2, 5) \).

3. **Substituting the Values:**
- For \( x \):
\[ x = \frac{1 \cdot 4 - 2 \cdot (-1)}{1 - 2} = \frac{4 + 2}{-1} = \frac{6}{-1} = -6 \]
- For \( y \):
\[ y = \frac{1 \cdot (-2) - 2 \cdot 2}{1 - 2} = \frac{-2 - 4}{-1} = \frac{-6}{-1} = 6 \]
- For \( z \):
\[ z = \frac{1 \cdot 5 - 2 \cdot 3}{1 - 2} = \frac{5 - 6}{-1} = \frac{-1}{-1} = 1 \]

4. **Final Coordinates:**
- Therefore, the coordinates of the point that divides the line segment externally in the ratio 1:2 are:
\[ (-6, 6, 1) \]

### Summary:
The required coordinates are \( (-6, 6, 1) \).