Find the ratio in which the line segment joining A(1,-5) and B(-4,5) is divided by the x - axis . Also find the co - ordinates of the point of division.

Answer

To find the ratio in which the line segment joining points A(1, -5) and B(-4, 5) is divided by the x-axis, we can follow these steps:

### Step 1: Understand the problem
The x-axis is represented by the equation y = 0. We need to find the point on the x-axis that divides the line segment AB in a certain ratio.

### 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:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
In our case, we need to find the coordinates of point P on the x-axis, which means we want the y-coordinate to be 0.

### Step 3: Set up the equation
Let the ratio in which the x-axis divides the segment AB be m:n. The coordinates of point P can be expressed as:
\[ P\left(\frac{m(-4) + n(1)}{m+n}, \frac{m(5) + n(-5)}{m+n}\right) \]
Since P lies on the x-axis, we set the y-coordinate to 0:
\[ \frac{m(5) + n(-5)}{m+n} = 0 \]

### Step 4: Solve for the ratio
From the equation above, we can simplify:
\[ m(5) + n(-5) = 0 \]
This implies:
\[ 5m - 5n = 0 \quad \Rightarrow \quad m = n \]
Thus, the ratio is 1:1.

### Step 5: Find the coordinates of the point of division
Now that we have the ratio, we can substitute m = n into the section formula to find the coordinates of point P:
\[ P\left(\frac{1(-4) + 1(1)}{1+1}, \frac{1(5) + 1(-5)}{1+1}\right) = P\left(\frac{-4 + 1}{2}, \frac{5 - 5}{2}\right) = P\left(\frac{-3}{2}, 0\right) \]

### Conclusion
The ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis is 1:1, and the coordinates of the point of division are \((-1.5, 0)\).