The ratio in which the line segment joining (2, -3) and (5,6) is divided by the x- axis is :

Options

A. `3 : 1 `
B. ` 1 : 2 `
C. ` sqrt(3) : 2 `
D. ` sqrt(2) : 3 `

Answer

To find the ratio in which the line segment joining the points (2, -3) and (5, 6) is divided by the x-axis, we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- \( A(2, -3) \) (which we can denote as \( (x_1, y_1) \))
- \( B(5, 6) \) (which we can denote as \( (x_2, y_2) \))

### Step 2: Set Up the Equation
The x-axis has the equation \( y = 0 \). We need to find the point \( P(x, 0) \) where the line segment \( AB \) intersects the x-axis. According to the section formula, the y-coordinate of point \( P \) can be expressed as:

\[ y = \frac{m y_2 + n y_1}{m + n} \]

where \( m \) and \( n \) are the ratios in which the segment is divided.

### Step 3: Substitute Known Values
Since \( P \) lies on the x-axis, we set \( y = 0 \). Thus, we have:

\[ 0 = \frac{m(6) + n(-3)}{m + n} \]

### Step 4: Simplify the Equation
Multiplying both sides by \( m + n \) (assuming \( m + n \neq 0 \)), we get:

\[ 0 = 6m - 3n \]

Rearranging gives:

\[ 6m = 3n \]

### Step 5: Find the Ratio
Dividing both sides by 3, we get:

\[ 2m = n \]

This can be rewritten as:

\[ \frac{m}{n} = \frac{1}{2} \]

### Step 6: Conclusion
Thus, the ratio in which the line segment joining the points (2, -3) and (5, 6) is divided by the x-axis is \( 1:2 \).