Find the ratio in which x - axis divides the join of A(2, - 3) and B(5, 6)

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: Understand the Problem
The x-axis divides the line segment at a point where the y-coordinate is 0. We need to find the ratio in which the x-axis divides the segment AB.

### 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 can be calculated as:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]

### Step 3: Assign Coordinates
Let the coordinates of A be (x1, y1) = (2, -3) and B be (x2, y2) = (5, 6). Let the ratio in which the x-axis divides the line segment be k:1.

### Step 4: Set Up the Equation
Since the y-coordinate of the point where the x-axis intersects is 0, we can set up the equation using the section formula for the y-coordinate:
\[ 0 = \frac{k \cdot 6 + 1 \cdot (-3)}{k + 1} \]

### Step 5: Solve for k
Now, we simplify the equation:
\[ 0 = \frac{6k - 3}{k + 1} \]
This implies:
\[ 6k - 3 = 0 \]
\[ 6k = 3 \]
\[ k = \frac{3}{6} = \frac{1}{2} \]

### Step 6: Determine the Ratio
The ratio in which the x-axis divides the segment AB is k:1, which is:
\[ \frac{1}{2}:1 = 1:2 \]

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

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