Find the coordinates of the point which divides the line segment joining the point, `(-2,3,5)` and `(1,-4,6)` in the ratio.
2 : 3 externally,

Answer

To find the coordinates of the point that divides the line segment joining the points \((-2, 3, 5)\) and \((1, -4, 6)\) in the ratio \(2:3\) externally, we can use the formula for external division in three-dimensional geometry.

### Step-by-Step Solution:

1. **Identify the Points and Ratios**:
Let the points be \(A(-2, 3, 5)\) and \(B(1, -4, 6)\). The ratio in which the line segment is divided is \(m_1:m_2 = 2:3\).

2. **Use the External Division Formula**:
The coordinates of the point \(P(x, y, z)\) that divides the line segment externally in the ratio \(m_1:m_2\) can be calculated using the following formulas:
\[ x = \frac{m_1 x_2 - m_2 x_1}{m_1 - m_2} \]
\[ y = \frac{m_1 y_2 - m_2 y_1}{m_1 - m_2} \]
\[ z = \frac{m_1 z_2 - m_2 z_1}{m_1 - m_2} \]
Here, \(m_1 = 2\), \(m_2 = 3\), \(A(x_1, y_1, z_1) = (-2, 3, 5)\), and \(B(x_2, y_2, z_2) = (1, -4, 6)\).

3. **Calculate the x-coordinate**:
\[ x = \frac{2 \cdot 1 - 3 \cdot (-2)}{2 - 3} \]
\[ = \frac{2 + 6}{-1} = \frac{8}{-1} = -8 \]

4. **Calculate the y-coordinate**:
\[ y = \frac{2 \cdot (-4) - 3 \cdot 3}{2 - 3} \]
\[ = \frac{-8 - 9}{-1} = \frac{-17}{-1} = 17 \]

5. **Calculate the z-coordinate**:
\[ z = \frac{2 \cdot 6 - 3 \cdot 5}{2 - 3} \]
\[ = \frac{12 - 15}{-1} = \frac{-3}{-1} = 3 \]

6. **Final Coordinates**:
The coordinates of the point \(P\) that divides the line segment externally in the ratio \(2:3\) are:
\[ P(-8, 17, 3) \]

### Summary of the Solution:
The coordinates of the point that divides the line segment joining the points \((-2, 3, 5)\) and \((1, -4, 6)\) in the ratio \(2:3\) externally are \((-8, 17, 3)\).