In what ratio does the x-axis divide the join of A(2, -3) and B(5, 6)?

Options

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

Explanation

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: Understand the problem
We need to find the point where the x-axis intersects the line segment joining points A and B. The x-axis is represented by the equation y = 0.

### Step 2: Use the section formula
The coordinates of the point that divides the line segment joining two points A(x1, y1) and B(x2, y2) in the ratio m:n is given by:
\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In our case, we will let the ratio be k:1, where k is the ratio we need to find.

### Step 3: Assign coordinates
Given:
- A(2, -3) → (x1, y1) = (2, -3)
- B(5, 6) → (x2, y2) = (5, 6)

### Step 4: Set up the coordinates of point C
Using the section formula, the coordinates of point C that divides AB in the ratio k:1 are:
\[ C = \left( \frac{k \cdot 5 + 1 \cdot 2}{k + 1}, \frac{k \cdot 6 + 1 \cdot (-3)}{k + 1} \right) \]

### Step 5: Set the y-coordinate to 0
Since point C lies on the x-axis, the y-coordinate must be 0:
\[ \frac{k \cdot 6 - 3}{k + 1} = 0 \]

### Step 6: Solve for k
Setting the numerator equal to zero:
\[ k \cdot 6 - 3 = 0 \]
\[ 6k = 3 \]
\[ k = \frac{3}{6} = \frac{1}{2} \]

### Step 7: Write the final ratio
The ratio in which the x-axis divides the line segment joining A and B is:
\[ 1:2 \]

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

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