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 internally,

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 \(2:3\) internally, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the Points and the Ratio**:
- Let point \(A\) be \((-2, 3, 5)\) and point \(B\) be \((1, -4, 6)\).
- The ratio in which the line segment is divided is \(2:3\). Here, \(M = 2\) and \(N = 3\).

2. **Use the Section Formula**:
The coordinates of the point \(C\) that divides the line segment \(AB\) in the ratio \(M:N\) are given by:
\[ C\left( \frac{M x_2 + N x_1}{M + N}, \frac{M y_2 + N y_1}{M + N}, \frac{M z_2 + N z_1}{M + N} \right) \]
where \((x_1, y_1, z_1)\) are the coordinates of point \(A\) and \((x_2, y_2, z_2)\) are the coordinates of point \(B\).

3. **Substitute the Values**:
- For the \(x\)-coordinate:
\[ x = \frac{2 \cdot 1 + 3 \cdot (-2)}{2 + 3} = \frac{2 - 6}{5} = \frac{-4}{5} \]
- For the \(y\)-coordinate:
\[ y = \frac{2 \cdot (-4) + 3 \cdot 3}{2 + 3} = \frac{-8 + 9}{5} = \frac{1}{5} \]
- For the \(z\)-coordinate:
\[ z = \frac{2 \cdot 6 + 3 \cdot 5}{2 + 3} = \frac{12 + 15}{5} = \frac{27}{5} \]

4. **Combine the Coordinates**:
Therefore, the coordinates of point \(C\) are:
\[ C\left(-\frac{4}{5}, \frac{1}{5}, \frac{27}{5}\right) \]

### Final Answer:
The coordinates of the point that divides the line segment joining the points \((-2, 3, 5)\) and \((1, -4, 6)\) in the ratio \(2:3\) internally are:
\[ \left(-\frac{4}{5}, \frac{1}{5}, \frac{27}{5}\right) \]