Find the co - ordinates of the point which divides the join `(1, 7) and (4,-3)` in the ratio `2:3`.

Answer

To find the coordinates of the point that divides the line segment joining the points \( A(1, 7) \) and \( B(4, -3) \) in the ratio \( 2:3 \), we can use the section formula. The section formula states that if a point \( P(x, y) \) divides the line segment joining the points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m_1:m_2 \), then the coordinates of point \( P \) can be calculated using the following formulas:

\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]

### Step-by-step solution:

1. **Identify the coordinates of points A and B**:
- \( A(x_1, y_1) = (1, 7) \)
- \( B(x_2, y_2) = (4, -3) \)

2. **Identify the ratio**:
- The ratio in which the point divides the line segment is \( m_1:m_2 = 2:3 \).
- Here, \( m_1 = 2 \) and \( m_2 = 3 \).

3. **Calculate the x-coordinate of point P**:
\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} = \frac{2 \cdot 4 + 3 \cdot 1}{2 + 3} \]
\[ x = \frac{8 + 3}{5} = \frac{11}{5} \]

4. **Calculate the y-coordinate of point P**:
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} = \frac{2 \cdot (-3) + 3 \cdot 7}{2 + 3} \]
\[ y = \frac{-6 + 21}{5} = \frac{15}{5} = 3 \]

5. **Combine the results**:
- The coordinates of the point \( P \) that divides the segment in the ratio \( 2:3 \) are:
\[ P\left(\frac{11}{5}, 3\right) \]

### Final Answer:
The coordinates of the point that divides the line segment joining \( (1, 7) \) and \( (4, -3) \) in the ratio \( 2:3 \) are \( \left(\frac{11}{5}, 3\right) \).