Given a line segment AB joining the points A (-4, 6) and B (8, -3). Find :
(i) the ratio in which AB is dividend by the Y-axis.

Answer

To find the ratio in which the line segment AB, joining the points A(-4, 6) and B(8, -3), is divided by the Y-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**:
- Point A has coordinates (-4, 6).
- Point B has coordinates (8, -3).

2. **Determine the Point of Intersection with the Y-axis**:
- The Y-axis is represented by the line where x = 0.
- We need to find the coordinates of the point P (0, y) where the line segment AB intersects the Y-axis.

3. **Use 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 will let the ratio be k:1. Thus, m = k and n = 1.

4. **Set Up the Equation for x-coordinate**:
- Since P lies on the Y-axis, we set the x-coordinate of P to 0:
\[ 0 = \frac{k \cdot 8 + 1 \cdot (-4)}{k + 1} \]

5. **Solve for k**:
- Multiplying both sides by (k + 1):
\[ 0 = 8k - 4 \]
- Rearranging gives:
\[ 8k = 4 \quad \Rightarrow \quad k = \frac{4}{8} = \frac{1}{2} \]

6. **Determine the Ratio**:
- The ratio in which AB is divided by the Y-axis is k:1, which is:
\[ \frac{1}{2}:1 \quad \Rightarrow \quad 1:2 \]

### Final Answer:
The ratio in which the line segment AB is divided by the Y-axis is **1:2**.