Find the ratio in which x-axis divides the join of points (2, -3) and (5,6) internally.

Answer

To find the ratio in which the x-axis divides the line segment joining the points \( A(2, -3) \) and \( B(5, 6) \) internally, we will use the concept of internal division in coordinate geometry.

### Step-by-step Solution:

1. **Identify the Points**:
We have two points:
- \( A(2, -3) \)
- \( B(5, 6) \)

2. **Set Up the Internal Division Formula**:
The internal division formula states that if a point \( P(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m_1 : m_2 \), then:
\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]

3. **Determine the Coordinates of the Point on the x-axis**:
Since the x-axis has a y-coordinate of 0, we set \( y = 0 \).

4. **Substituting the Values**:
Using the coordinates of points \( A \) and \( B \):
- \( x_1 = 2, y_1 = -3 \)
- \( x_2 = 5, y_2 = 6 \)
We substitute these values into the y-coordinate equation:
\[ 0 = \frac{m_1 \cdot 6 + m_2 \cdot (-3)}{m_1 + m_2} \]

5. **Simplifying the Equation**:
This leads to:
\[ 0 = 6m_1 - 3m_2 \]
Rearranging gives:
\[ 6m_1 = 3m_2 \]

6. **Finding the Ratio**:
Dividing both sides by 3:
\[ 2m_1 = m_2 \]
This implies:
\[ \frac{m_1}{m_2} = \frac{1}{2} \]
Therefore, the ratio \( m_1 : m_2 = 1 : 2 \).

### Final Answer:
The x-axis divides the line segment joining the points \( (2, -3) \) and \( (5, 6) \) in the ratio \( 1 : 2 \).