Find the ratio in which Y-axis divides the line segment joining the points (a, b) and (-b, a).

Answer

To find the ratio in which the Y-axis divides the line segment joining the points (a, b) and (-b, a), we can follow these steps:

### Step 1: Identify the Points
We have two points:
- Point A: (a, b)
- Point B: (-b, a)

### Step 2: Understand the Y-axis
The Y-axis is represented by the equation x = 0. Any point on the Y-axis will have its x-coordinate as 0, and we can denote the point where the Y-axis intersects the line segment as P(0, y).

### Step 3: Assume the Ratio
Let the Y-axis divide the line segment AB in the ratio λ:1. This means we can express the coordinates of point P using the section formula.

### Step 4: Apply the Section Formula
According to the section formula, the coordinates of point P (which divides the line segment joining points A and B in the ratio λ:1) can be calculated as follows:

\[ P_x = \frac{(x_1 \cdot m + x_2 \cdot n)}{m+n} \]
\[ P_y = \frac{(y_1 \cdot m + y_2 \cdot n)}{m+n} \]

Where:
- \( (x_1, y_1) = (a, b) \)
- \( (x_2, y_2) = (-b, a) \)
- \( m = λ \) and \( n = 1 \)

### Step 5: Set Up the Equation for X-coordinate
Since P lies on the Y-axis, we have:
\[ P_x = 0 \]
Thus, we can set up the equation:
\[ 0 = \frac{(a \cdot 1 + (-b) \cdot λ)}{λ + 1} \]

### Step 6: Solve for λ
Multiplying both sides by (λ + 1) gives:
\[ 0 = a - bλ \]
Rearranging this, we find:
\[ bλ = a \quad \Rightarrow \quad λ = \frac{a}{b} \]

### Step 7: Write the Ratio
Since we assumed that the ratio is λ:1, we can express the ratio as:
\[ \text{Ratio} = \frac{λ}{1} = \frac{a}{b} \]

### Conclusion
Thus, the ratio in which the Y-axis divides the line segment joining the points (a, b) and (-b, a) is:
\[ \frac{a}{b} \]