x-axis divides the line segment joining A(2, -3) and B(5,6) in the ratio :

Options

A. `2:3`
B. `3:5 `
C. `1:2`
D. `2:1`

Answer

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

### Step 1: Identify the coordinates of points A and B.
- A(2, -3)
- B(5, 6)

### Step 2: Understand that the x-axis has a y-coordinate of 0.
- The point where the x-axis intersects the line segment will have coordinates (x, 0).

### Step 3: Use the section formula.
The section formula states that if a point P divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m1:m2, then the coordinates of point P are given by:
\[ P\left(\frac{m1 \cdot x2 + m2 \cdot x1}{m1 + m2}, \frac{m1 \cdot y2 + m2 \cdot y1}{m1 + m2}\right) \]

### Step 4: Set the y-coordinate of the dividing point to 0.
Since we want the y-coordinate to be 0 (because it lies on the x-axis), we set up the equation:
\[ \frac{m1 \cdot 6 + m2 \cdot (-3)}{m1 + m2} = 0 \]

### Step 5: Solve the equation for the ratio.
Multiplying both sides by (m1 + m2) gives:
\[ m1 \cdot 6 + m2 \cdot (-3) = 0 \]
This simplifies to:
\[ 6m1 = 3m2 \]
Dividing both sides by 3 gives:
\[ 2m1 = m2 \]
Thus, we can express the ratio:
\[ \frac{m1}{m2} = \frac{1}{2} \]

### Step 6: Write the final answer.
The x-axis divides the line segment joining A(2, -3) and B(5, 6) in the ratio **1:2**.

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