Find the ratio in which the line joining `(5, -6) and (-1, -4)` is divided by x - axis. Also find the coordinates of the point of intersection.

Answer

To solve the problem, we need to find the ratio in which the line joining the points \( A(5, -6) \) and \( B(-1, -4) \) is divided by the x-axis, and also determine the coordinates of the point of intersection.

### Step 1: Identify the Points
Let point \( A \) be \( (5, -6) \) and point \( B \) be \( (-1, -4) \). The x-axis has the equation \( y = 0 \).

### Step 2: Use the Section Formula
The section formula states that if a point \( P(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( P \) can be given by:
\[ P\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Since point \( P \) lies on the x-axis, its y-coordinate is \( 0 \). Therefore, we can set up the equation using the y-coordinates of points \( A \) and \( B \).

### Step 3: Set Up the Equation for y-coordinate
Let the ratio in which point \( P \) divides the line segment be \( m:n \). The y-coordinate of point \( P \) can be expressed as:
\[ \frac{m(-4) + n(-6)}{m+n} = 0 \]

### Step 4: Solve for the Ratio
Rearranging the equation gives:
\[ m(-4) + n(-6) = 0 \]
This simplifies to:
\[ -4m - 6n = 0 \]
or
\[ 4m = -6n \]
This can be rearranged to find the ratio:
\[ \frac{m}{n} = \frac{6}{4} = \frac{3}{2} \]

### Step 5: Conclusion on the Ratio
Thus, the line segment joining points \( A \) and \( B \) is divided by the x-axis in the ratio \( 3:2 \).

### Step 6: Find the Coordinates of Point P
Now we can find the coordinates of point \( P \) using the section formula. We can substitute \( m = 3 \) and \( n = 2 \):
\[ P\left( \frac{3(-1) + 2(5)}{3+2}, \frac{3(-4) + 2(-6)}{3+2} \right) \]
Calculating the x-coordinate:
\[ P_x = \frac{-3 + 10}{5} = \frac{7}{5} \]
Calculating the y-coordinate:
\[ P_y = \frac{-12 - 12}{5} = \frac{-24}{5} \]
Since we are looking for the intersection with the x-axis, we set \( P_y = 0 \).

### Final Coordinates of Point P
Thus, the coordinates of point \( P \) where the line intersects the x-axis are:
\[ P\left( \frac{7}{5}, 0 \right) \]

### Summary of Results
- The ratio in which the line joining the points \( (5, -6) \) and \( (-1, -4) \) is divided by the x-axis is \( 3:2 \).
- The coordinates of the point of intersection are \( \left( \frac{7}{5}, 0 \right) \).