Find the ratio in which Y-axis divides the line segment joining the points (3, 4) and (-2, 5).

Answer

To find the ratio in which the Y-axis divides the line segment joining the points (3, 4) and (-2, 5), we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- \( A(3, 4) \)
- \( B(-2, 5) \)

### Step 2: Determine the Point on the Y-axis
Since the Y-axis has the equation \( x = 0 \), we need to find the point \( P(0, y) \) that divides the line segment \( AB \).

### Step 3: Use the Section Formula
The section formula states that if a point \( P(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step 4: Set Up the Equation for the X-coordinate
Since \( P \) lies on the Y-axis, we set \( x = 0 \):
\[ 0 = \frac{m(-2) + n(3)}{m+n} \]

### Step 5: Solve for the Ratio
From the equation above, we can multiply both sides by \( m+n \) (assuming \( m+n \neq 0 \)):
\[ 0 = m(-2) + n(3) \]
This simplifies to:
\[ -2m + 3n = 0 \]
Rearranging gives:
\[ 2m = 3n \quad \Rightarrow \quad \frac{m}{n} = \frac{3}{2} \]

### Step 6: Write the Ratio
Thus, the ratio in which the Y-axis divides the line segment joining the points \( (3, 4) \) and \( (-2, 5) \) is:
\[ m:n = 3:2 \]