Write the coordinates ot the point dividing line segment joining the points (1, 5) and (-5,1) internally in the ratio 1: 4.

Answer

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

### Step-by-step Solution:

1. **Identify the Points and the Ratio**:
- Let point P be (1, 5) and point Q be (-5, 1).
- The ratio in which the point R divides the line segment PQ is 1:4.

2. **Assign Variables**:
- Let \( m_1 = 1 \) (the part of the ratio closer to point P) and \( m_2 = 4 \) (the part of the ratio closer to point Q).

3. **Use the Section Formula**:
The section formula for finding the coordinates of a point dividing a line segment internally in the ratio \( m_1:m_2 \) is given by:
\[ R(x, y) = \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right) \]
Here, \( (x_1, y_1) \) are the coordinates of point P and \( (x_2, y_2) \) are the coordinates of point Q.

4. **Substitute the Values**:
- For point P: \( x_1 = 1 \), \( y_1 = 5 \)
- For point Q: \( x_2 = -5 \), \( y_2 = 1 \)
- Substitute into the formula:
\[ R_x = \frac{1 \cdot (-5) + 4 \cdot 1}{1 + 4} = \frac{-5 + 4}{5} = \frac{-1}{5} \]
\[ R_y = \frac{1 \cdot 1 + 4 \cdot 5}{1 + 4} = \frac{1 + 20}{5} = \frac{21}{5} \]

5. **Final Coordinates**:
- Thus, the coordinates of point R are:
\[ R = \left( -\frac{1}{5}, \frac{21}{5} \right) \]

### Final Answer:
The coordinates of the point dividing the line segment joining (1, 5) and (-5, 1) in the ratio 1:4 are \( \left( -\frac{1}{5}, \frac{21}{5} \right) \).