In what ratio is the line segment joining the points A(-2, -3) and B(3, 7) divided by the y-axis? Also, find the coordinates of the point of division.

Answer

To solve the problem of finding the ratio in which the line segment joining the points A(-2, -3) and B(3, 7) is divided by the y-axis, and to find the coordinates of the point of division, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**:
- Let A = (-2, -3) and B = (3, 7).

2. **Understanding the Y-axis**:
- The y-axis is represented by the line x = 0. We need to find the point on the y-axis where the line segment AB intersects.

3. **Using the Section Formula**:
- 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:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
- Here, we want the x-coordinate of P to be 0 (since it lies on the y-axis).

4. **Setting Up the Equation**:
- Let the ratio in which the y-axis divides the segment AB be k:1. Thus, we have:
\[ P\left(\frac{k \cdot 3 + 1 \cdot (-2)}{k + 1}, \frac{k \cdot 7 + 1 \cdot (-3)}{k + 1}\right) \]
- Setting the x-coordinate to 0 gives:
\[ \frac{3k - 2}{k + 1} = 0 \]

5. **Solving for k**:
- From the equation \(3k - 2 = 0\):
\[ 3k = 2 \implies k = \frac{2}{3} \]

6. **Finding the Coordinates of the Point of Division**:
- Now substitute k back into the y-coordinate formula:
\[ y = \frac{7k - 3}{k + 1} \]
- Substitute \(k = \frac{2}{3}\):
\[ y = \frac{7 \cdot \frac{2}{3} - 3}{\frac{2}{3} + 1} = \frac{\frac{14}{3} - 3}{\frac{2}{3} + \frac{3}{3}} = \frac{\frac{14}{3} - \frac{9}{3}}{\frac{5}{3}} = \frac{\frac{5}{3}}{\frac{5}{3}} = 1 \]

7. **Final Coordinates and Ratio**:
- The point of division is (0, 1).
- The ratio in which the line segment is divided by the y-axis is \( \frac{2}{3} : 1 \) or simply 2:3.

### Final Answer:
- The ratio in which the line segment joining A and B is divided by the y-axis is 2:3.
- The coordinates of the point of division are (0, 1).