P(1, -2) is a point on the line segment A(3, -6) and B(x, y) such that `AP:PB` is equal to `2:3`. Find the coordinates of B.

Answer

To find the coordinates of point B given that point P(1, -2) divides the line segment AB in the ratio 2:3, we can use the section formula.

### Step-by-Step Solution:

1. **Identify the points and the ratio**:
- Let A(3, -6) and B(x, y).
- The point P(1, -2) divides the line segment AB in the ratio 2:3.

2. **Use the section formula for the x-coordinates**:
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, \(m = 2\), \(n = 3\), \(x_1 = 3\) (x-coordinate of A), and \(P_x = 1\).

Plugging in the values:
\[ 1 = \frac{2 \cdot x + 3 \cdot 3}{2 + 3} \]
\[ 1 = \frac{2x + 9}{5} \]

3. **Solve for x**:
Multiply both sides by 5:
\[ 5 = 2x + 9 \]
Rearranging gives:
\[ 2x = 5 - 9 \]
\[ 2x = -4 \]
\[ x = -2 \]

4. **Use the section formula for the y-coordinates**:
Similarly, for the y-coordinates:
\[ P_y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \]
Here, \(y_1 = -6\) and \(P_y = -2\).

Plugging in the values:
\[ -2 = \frac{2 \cdot y + 3 \cdot (-6)}{2 + 3} \]
\[ -2 = \frac{2y - 18}{5} \]

5. **Solve for y**:
Multiply both sides by 5:
\[ -10 = 2y - 18 \]
Rearranging gives:
\[ 2y = -10 + 18 \]
\[ 2y = 8 \]
\[ y = 4 \]

6. **Final coordinates of B**:
Therefore, the coordinates of point B are:
\[ B\left(-2, 4\right) \]

### Summary:
The coordinates of point B are \((-2, 4)\).