In which ratio X Axis will divide the join of two point `(3,-6),(4,5)`

Answer

To find the ratio in which the X-axis divides the line segment joining the points \( A(3, -6) \) and \( B(4, 5) \), we will follow these steps:

### Step 1: Understand the Problem
We need to find the ratio in which the X-axis (which has a y-coordinate of 0) divides the line segment joining the points \( A(3, -6) \) and \( B(4, 5) \).

### Step 2: Set Up the Equation
Let the ratio in which the X-axis divides the segment \( AB \) be \( m:n \). The coordinates of the point dividing the segment in the ratio \( m:n \) can be found using the section formula:

\[ C\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Here, \( (x_1, y_1) = (3, -6) \) and \( (x_2, y_2) = (4, 5) \).

### Step 3: Apply the Section Formula
We need to find the y-coordinate of point \( C \) when it lies on the X-axis, which is \( 0 \).

Using the section formula for the y-coordinate:

\[ y_C = \frac{m \cdot 5 + n \cdot (-6)}{m+n} = 0 \]

### Step 4: Solve the Equation
Setting the equation to zero gives:

\[ m \cdot 5 - n \cdot 6 = 0 \]

Rearranging this, we have:

\[ 5m = 6n \]

### Step 5: Find the Ratio
From the equation \( 5m = 6n \), we can express the ratio \( \frac{m}{n} \):

\[ \frac{m}{n} = \frac{6}{5} \]

Thus, the ratio \( m:n = 6:5 \).

### Conclusion
The X-axis divides the line segment joining the points \( (3, -6) \) and \( (4, 5) \) in the ratio \( 6:5 \).

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