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

Answer

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

### Step-by-Step Solution:

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

2. **Use the Section Formula**:
The coordinates \( (x, y) \) of the point that divides the line segment joining the points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) can be calculated using the formulas:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

3. **Substitute the Values**:
- Here, \( x_1 = -1, y_1 = 2, x_2 = 3, y_2 = 5, m = 3, n = 5 \).
- Substitute these values into the formulas.

For \( x \):
\[ x = \frac{3 \cdot 3 + 5 \cdot (-1)}{3 + 5} = \frac{9 - 5}{8} = \frac{4}{8} = \frac{1}{2} \]

For \( y \):
\[ y = \frac{3 \cdot 5 + 5 \cdot 2}{3 + 5} = \frac{15 + 10}{8} = \frac{25}{8} \]

4. **Final Coordinates**:
- The coordinates of the point that divides the line segment in the ratio 3:5 are:
\[ \left( \frac{1}{2}, \frac{25}{8} \right) \]

### Final Answer:
The coordinates of the required point are \( \left( \frac{1}{2}, \frac{25}{8} \right) \).

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