In what ratio is the line joining the points (4,5) and (1,2) is divided by X-axis

Options

A. 2 : 5 externally
B. 2: 5 internally
C. 3 : 4 externally
D. 3: 4 internally

Answer

To solve the problem of finding the ratio in which the line joining the points (4, 5) and (1, 2) is divided by the X-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Points**:
Let A = (4, 5) and B = (1, 2). We need to find the point C where the line AB intersects the X-axis.

2. **Determine the Coordinates of Point C**:
Since point C lies on the X-axis, its y-coordinate is 0. Therefore, we can denote point C as (x, 0).

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 can be given by:
\[ C = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
Here, we will let the ratio be K:1, where K is the unknown we need to find.

4. **Substituting the Values**:
- For point A (4, 5): \(x_1 = 4\) and \(y_1 = 5\)
- For point B (1, 2): \(x_2 = 1\) and \(y_2 = 2\)
- The coordinates of C can then be expressed as:
\[ C = \left( \frac{K \cdot 1 + 1 \cdot 4}{K + 1}, \frac{K \cdot 2 + 1 \cdot 5}{K + 1} \right) \]

5. **Setting the y-coordinate to 0**:
Since C lies on the X-axis, we set the y-coordinate to 0:
\[ \frac{K \cdot 2 + 1 \cdot 5}{K + 1} = 0 \]

6. **Solving for K**:
To solve for K, we can set the numerator equal to zero:
\[ K \cdot 2 + 5 = 0 \]
\[ 2K + 5 = 0 \implies 2K = -5 \implies K = -\frac{5}{2} \]

7. **Interpreting the Ratio**:
The ratio in which the line is divided by the X-axis is K:1, which translates to:
\[ -\frac{5}{2} : 1 \]
This implies that the line is divided externally in the ratio of 2:5.

### Final Answer:
The line joining the points (4, 5) and (1, 2) is divided by the X-axis in the ratio of 2:5 externally.