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 use the section formula. Here’s a step-by-step solution:

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

### Step 2: Determine the Coordinates of the Point on the Y-axis
The point where the Y-axis divides the line segment will have coordinates of the form \( (0, b) \), where \( b \) is some value we need to find.

### Step 3: Use the Section Formula
According to the section formula, 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 the coordinates of point \( P \) can be given by:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

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

### Step 5: Cross Multiply
Cross-multiplying gives us:
\[ 0 = -2m + 3n \]
This simplifies to:
\[ 2m = 3n \]

### Step 6: Find the Ratio
From the equation \( 2m = 3n \), we can express the ratio \( \frac{m}{n} \):
\[ \frac{m}{n} = \frac{3}{2} \]
Thus, the ratio in which the Y-axis divides the line segment joining the points (3, 4) and (-2, 5) is \( 3:2 \).

### Final Answer
The Y-axis divides the line segment joining the points (3, 4) and (-2, 5) in the ratio \( 3:2 \).

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