If the point P `(-1,2)` divides the line segment joining A (2,5) and B in the ratio `3 : 4` , find the co-ordinate of B .

Answer

To find the coordinates of point B given that point P divides the line segment joining A and B in the ratio 3:4, we will use the section formula.

### Step-by-Step Solution:

1. **Identify the given points and ratio**:
- Point A = (2, 5)
- Point P = (-1, 2)
- Let the coordinates of point B be (x, y).
- The ratio in which P divides AB = 3:4.

2. **Use the section formula for the x-coordinate**:
The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of P are given by:
\[ P_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \]
Here, \(m = 3\), \(n = 4\), \(x_1 = 2\), and \(P_x = -1\).

Plugging in the values:
\[ -1 = \frac{3x + 4 \cdot 2}{3 + 4} \]
\[ -1 = \frac{3x + 8}{7} \]

3. **Cross-multiply to solve for x**:
\[ -7 = 3x + 8 \]
\[ 3x = -7 - 8 \]
\[ 3x = -15 \]
\[ x = -5 \]

4. **Use the section formula for the y-coordinate**:
Similarly, for the y-coordinate:
\[ P_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \]
Here, \(y_1 = 5\), \(y_2 = y\), and \(P_y = 2\).

Plugging in the values:
\[ 2 = \frac{3y + 4 \cdot 5}{3 + 4} \]
\[ 2 = \frac{3y + 20}{7} \]

5. **Cross-multiply to solve for y**:
\[ 14 = 3y + 20 \]
\[ 3y = 14 - 20 \]
\[ 3y = -6 \]
\[ y = -2 \]

6. **Conclusion**:
The coordinates of point B are \((-5, -2)\).

### Final Answer:
The coordinates of point B are \((-5, -2)\).