Find the ratio in which the line segment joining the points (1, -3) and (4 , 5) is divided by x- axis . Also find the co-ordinates of this point on x-axis.

Answer

To solve the problem of finding the ratio in which the line segment joining the points (1, -3) and (4, 5) is divided by the x-axis, and to find the coordinates of the point on the x-axis, we can follow these steps:

### Step 1: Identify the Points
The two points given are:
- Point A (1, -3)
- Point B (4, 5)

### Step 2: Set Up the Problem
The point on the x-axis will have coordinates (x, 0) because the y-coordinate of any point on the x-axis is 0. We need to find the ratio in which this point divides the line segment AB.

### Step 3: Use the Section Formula
The section formula states that if a point divides a line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the point (x, y) are given by:

\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

In our case, we can let the ratio be k:1 (where k is the unknown we want to find). Therefore, we have:
- \(x_1 = 1\), \(y_1 = -3\)
- \(x_2 = 4\), \(y_2 = 5\)

### Step 4: Set Up the Equation for y-coordinate
Since the point lies on the x-axis, we set the y-coordinate to 0:

\[ 0 = \frac{k \cdot 5 + 1 \cdot (-3)}{k + 1} \]

### Step 5: Solve for k
Multiplying both sides by (k + 1) gives:

\[ 0 = k \cdot 5 - 3 \]

Rearranging gives:

\[ 5k = 3 \implies k = \frac{3}{5} \]

### Step 6: Find the Ratio
The ratio in which the line segment is divided is \(k:1\), which is:

\[ \frac{3}{5}:1 \implies 3:5 \]

### Step 7: Find the x-coordinate of the Point
Now, we can find the x-coordinate using the section formula:

\[ x = \frac{k \cdot x_2 + 1 \cdot x_1}{k + 1} \]

Substituting the values we have:

\[ x = \frac{\frac{3}{5} \cdot 4 + 1 \cdot 1}{\frac{3}{5} + 1} \]

Calculating the numerator:

\[ = \frac{\frac{12}{5} + 1}{\frac{3}{5} + 1} = \frac{\frac{12}{5} + \frac{5}{5}}{\frac{3}{5} + \frac{5}{5}} = \frac{\frac{17}{5}}{\frac{8}{5}} = \frac{17}{8} \]

### Step 8: Write the Coordinates
Thus, the coordinates of the point on the x-axis that divides the line segment in the ratio 3:5 are:

\[ \left(\frac{17}{8}, 0\right) \]

### Final Answer
The ratio in which the line segment is divided is **3:5**, and the coordinates of the point on the x-axis are **\(\left(\frac{17}{8}, 0\right)\)**.

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