In what ratio is the joining the points (4,2) and (3,-5) divided by the x-axia ? Also, find the co-ordinates of the point of intersection.

Answer

To solve the problem of finding the ratio in which the line joining the points (4, 2) and (3, -5) is divided by the x-axis, and to find the coordinates of the point of intersection, we can follow these steps:

### Step 1: Identify the Points
Let the points be:
- A (4, 2)
- B (3, -5)

### Step 2: Determine the Coordinates of the Intersection Point
Since the point of intersection (C) lies on the x-axis, its coordinates will be of the form (x, 0).

### Step 3: Use the Section Formula
The section formula states that if a point C divides the line segment joining points A (x1, y1) and B (x2, y2) in the ratio m:n, then the coordinates of point C are given by:
\[ C = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In this case, let the ratio be K:1. Therefore, we have:
- \(x_1 = 4\), \(y_1 = 2\)
- \(x_2 = 3\), \(y_2 = -5\)

### Step 4: Set Up the Coordinates of C
Using the section formula, the coordinates of point C will be:
\[ C = \left( \frac{3K + 4}{K + 1}, \frac{-5K + 2}{K + 1} \right) \]
Since C lies on the x-axis, the y-coordinate must be 0:
\[ \frac{-5K + 2}{K + 1} = 0 \]

### Step 5: Solve for K
Setting the numerator equal to zero gives:
\[ -5K + 2 = 0 \]
Solving for K:
\[ 5K = 2 \implies K = \frac{2}{5} \]

### Step 6: Determine the Ratio
The ratio in which the points are divided is K:1, which is:
\[ \frac{2}{5}:1 \implies 2:5 \]

### Step 7: Find the x-coordinate of Point C
Now we can find the x-coordinate of point C using the value of K:
\[ x = \frac{3K + 4}{K + 1} \]
Substituting \(K = \frac{2}{5}\):
\[ x = \frac{3 \cdot \frac{2}{5} + 4}{\frac{2}{5} + 1} = \frac{\frac{6}{5} + 4}{\frac{2}{5} + \frac{5}{5}} = \frac{\frac{6}{5} + \frac{20}{5}}{\frac{7}{5}} = \frac{\frac{26}{5}}{\frac{7}{5}} = \frac{26}{7} \]

### Step 8: Write the Coordinates of Point C
Thus, the coordinates of point C are:
\[ C = \left( \frac{26}{7}, 0 \right) \]

### Final Answer
The ratio in which the joining points (4, 2) and (3, -5) is divided by the x-axis is \(2:5\), and the coordinates of the point of intersection are \(\left( \frac{26}{7}, 0 \right)\).

---