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

Options

A. 5 : 2 externally
B. 3 : 4 internally
C. 3 : 4 externally
D. 1: 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 1: Identify the points
The two points given are A(4, 5) and B(1, 2).

### Step 2: Find the equation of the line
We can use the two-point form of the equation of a line, which is given by:

\[ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of points A(4, 5) and B(1, 2):

\[ \frac{y - 5}{x - 4} = \frac{2 - 5}{1 - 4} \]

This simplifies to:

\[ \frac{y - 5}{x - 4} = \frac{-3}{-3} = 1 \]

Cross-multiplying gives:

\[ y - 5 = x - 4 \]

Rearranging this, we find the equation of the line:

\[ y = x + 1 \]

### Step 3: Find the intersection with the X-axis
To find where this line intersects the X-axis, we set \(y = 0\):

\[ 0 = x + 1 \]

Solving for \(x\) gives:

\[ x = -1 \]

Thus, the point of intersection is \((-1, 0)\).

### Step 4: Use the section formula
Let the point \((-1, 0)\) divide the line segment AB in the ratio \(m: n\). According to the section formula, the coordinates of the point dividing the line segment in the ratio \(m:n\) are given by:

\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Substituting the coordinates of points A(4, 5) and B(1, 2):

\[ \left( \frac{m \cdot 1 + n \cdot 4}{m+n}, \frac{m \cdot 2 + n \cdot 5}{m+n} \right) = (-1, 0) \]

### Step 5: Set up equations
From the x-coordinates:

\[ \frac{m \cdot 1 + n \cdot 4}{m+n} = -1 \]

Cross-multiplying gives:

\[ m + 4n = -1(m + n) \]

This simplifies to:

\[ m + 4n = -m - n \implies 2m + 5n = 0 \tag{1} \]

From the y-coordinates:

\[ \frac{m \cdot 2 + n \cdot 5}{m+n} = 0 \]

Cross-multiplying gives:

\[ m \cdot 2 + n \cdot 5 = 0 \implies 2m + 5n = 0 \tag{2} \]

### Step 6: Solve the equations
From equation (2), we can express \(m\) in terms of \(n\):

\[ 2m = -5n \implies m = -\frac{5}{2}n \]

Substituting \(m\) into equation (1):

\[ 2\left(-\frac{5}{2}n\right) + 5n = 0 \]

This simplifies to:

\[ -5n + 5n = 0 \]

This confirms that our equations are consistent.

### Step 7: Determine the ratio
Since we have \(m = -\frac{5}{2}n\), the ratio \(m:n\) is:

\[ -\frac{5}{2}: 1 \]

This indicates that the point divides the line externally in the ratio \(5:2\).

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

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