Find the co-ordinates of point P which divides the line joining A (4,-5) and B (6,3) in the ratio 2 : 5.

Answer

To find the coordinates of point P that divides the line segment joining points A (4, -5) and B (6, 3) in the ratio 2:5, we will use the section formula.

### Step-by-Step Solution:

1. **Identify the coordinates of points A and B**:
- Point A (X1, Y1) = (4, -5)
- Point B (X2, Y2) = (6, 3)

2. **Identify the ratio in which point P divides the line segment**:
- The ratio is given as M:N = 2:5, where M = 2 and N = 5.

3. **Use the section formula to find the coordinates of point P**:
- The section formula states that if a point P divides the segment AB in the ratio M:N, then the coordinates (X, Y) of point P can be calculated as follows:
\[ X = \frac{M \cdot X2 + N \cdot X1}{M + N} \]
\[ Y = \frac{M \cdot Y2 + N \cdot Y1}{M + N} \]

4. **Substituting the values into the formula for X**:
- Substitute M = 2, N = 5, X1 = 4, and X2 = 6:
\[ X = \frac{2 \cdot 6 + 5 \cdot 4}{2 + 5} \]
\[ X = \frac{12 + 20}{7} = \frac{32}{7} \]

5. **Substituting the values into the formula for Y**:
- Substitute M = 2, N = 5, Y1 = -5, and Y2 = 3:
\[ Y = \frac{2 \cdot 3 + 5 \cdot (-5)}{2 + 5} \]
\[ Y = \frac{6 - 25}{7} = \frac{-19}{7} \]

6. **Final coordinates of point P**:
- Therefore, the coordinates of point P are:
\[ P\left(\frac{32}{7}, \frac{-19}{7}\right) \]

### Summary of the Solution:
The coordinates of point P that divides the line segment joining A (4, -5) and B (6, 3) in the ratio 2:5 are \( P\left(\frac{32}{7}, \frac{-19}{7}\right) \).