Find the ratio is which the line joining `A(1, -5) and B(-4,5)` 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 points A(1, -5) and B(-4, 5) is divided by the x-axis, and also find the coordinates of the point of intersection.

### Step 1: Understand the problem
We need to find the ratio in which the line segment AB is divided by the x-axis. The x-axis has a y-coordinate of 0, so we will denote the point of intersection as (x1, 0).

### 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 will let the ratio be λ:1, where λ is the unknown we need to find. Thus, we have:
- A(1, -5) → (x1, y1) = (1, -5)
- B(-4, 5) → (x2, y2) = (-4, 5)

### Step 3: Set up the equation for the y-coordinate
Since the point of intersection lies on the x-axis, the y-coordinate of point P is 0. Therefore, we can set up the equation for the y-coordinate:
\[ \frac{5λ + (-5) \cdot 1}{λ + 1} = 0 \]
This simplifies to:
\[ 5λ - 5 = 0 \]

### Step 4: Solve for λ
Now, we can solve for λ:
\[ 5λ = 5 \\ λ = 1 \]
Thus, the ratio in which the line is divided by the x-axis is 1:1.

### Step 5: Find the coordinates of the point of intersection
Since the ratio is 1:1, the point of intersection is the midpoint of line segment AB. We can find the midpoint using the midpoint formula:
\[ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]
Substituting the coordinates of A and B:
\[ \text{Midpoint} = \left(\frac{1 + (-4)}{2}, \frac{-5 + 5}{2}\right) = \left(\frac{-3}{2}, 0\right) \]
So, the coordinates of the point of intersection are \((-3/2, 0)\).

### Final Answer
The ratio in which the line joining A(1, -5) and B(-4, 5) is divided by the x-axis is **1:1**, and the coordinates of the point of intersection are **(-3/2, 0)**.