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 coordinates 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 of intersection of the line segment joining A and B with the X-axis and determine the ratio in which this point divides the segment.

### Step 2: Find the coordinates of the point of intersection
Let the point of intersection on the X-axis be P(x, 0). The coordinates of points A and B are A(1, -5) and B(-4, 5).

### Step 3: 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 want the y-coordinate of P to be 0 (since it lies on the X-axis). Thus, we can set up the equation for the y-coordinate:

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

### Step 4: Solve for the ratio
From the equation above, we can simplify:

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

This implies:

\[ m \cdot 5 = n \cdot 5 \]

Dividing both sides by 5 (assuming 5 is not zero):

\[ m = n \]

This means the ratio m:n is 1:1.

### Step 5: Find the coordinates of the point of division
Now, we can substitute m = 1 and n = 1 into the section formula to find the coordinates of point P:

\[ P\left(\frac{1 \cdot (-4) + 1 \cdot 1}{1 + 1}, \frac{1 \cdot 5 + 1 \cdot (-5)}{1 + 1}\right) \]

Calculating the x-coordinate:

\[ P_x = \frac{-4 + 1}{2} = \frac{-3}{2} = -1.5 \]

Calculating the y-coordinate:

\[ P_y = \frac{5 - 5}{2} = \frac{0}{2} = 0 \]

Thus, the coordinates of point P are (-1.5, 0).

### Final Result
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)**.

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