x -axis divides the join of (2, -3) and (5, 6) in the ratio _________.

Options

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

Answer

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

### Step 1: Identify the points
Let the points be:
- A (2, -3)
- B (5, 6)

### Step 2: 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 m:n, then the coordinates of point P are given by:
\[ P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

### Step 3: Set y-coordinate to 0
Since the x-axis divides the segment, we set the y-coordinate of point P to 0:
\[ 0 = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \]
Substituting the values of y1 and y2:
\[ 0 = \frac{m \cdot 6 + n \cdot (-3)}{m+n} \]

### Step 4: Simplify the equation
This simplifies to:
\[ m \cdot 6 - n \cdot 3 = 0 \]
Rearranging gives:
\[ 6m = 3n \]
Dividing both sides by 3:
\[ 2m = n \]
Thus, we can express n in terms of m:
\[ n = 2m \]

### Step 5: Express the ratio
The ratio in which the x-axis divides the line segment is:
\[ \frac{m}{n} = \frac{m}{2m} = \frac{1}{2} \]
This means the ratio is 1:2.

### Step 6: Conclusion
Thus, the x-axis divides the line segment joining the points (2, -3) and (5, 6) in the ratio **1:2**.

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