The ratio in which the line segment joining (2, -3) and (5,6) is divided by the x- axis is :

Options

A. `3 : 1 `
B. ` 1 : 2 `
C. ` sqrt(3) : 2 `
D. ` sqrt(2) : 3 `

Answer

To find the ratio in which the line segment joining the points (2, -3) and (5, 6) is divided by the x-axis, we can follow these steps:

### Step 1: Determine the coordinates of the points
The points given are A(2, -3) and B(5, 6).

### Step 2: Find the equation of the line joining the two points
To find the equation of the line, we first calculate the slope (m) of the line joining the points A and B.

The formula for the slope \( m \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates:
\[ m = \frac{6 - (-3)}{5 - 2} = \frac{6 + 3}{5 - 2} = \frac{9}{3} = 3 \]

Now, using the point-slope form of the line equation \( y - y_1 = m(x - x_1) \), we can use point A(2, -3):
\[ y - (-3) = 3(x - 2) \]
This simplifies to:
\[ y + 3 = 3x - 6 \]
\[ y = 3x - 9 \]

### Step 3: Find the intersection of the line with the x-axis
To find where the line intersects the x-axis, we set \( y = 0 \):
\[ 0 = 3x - 9 \]
Solving for \( x \):
\[ 3x = 9 \implies x = 3 \]
Thus, the intersection point is (3, 0).

### Step 4: Use the section formula to find the ratio
Let the ratio in which the x-axis divides the line segment AB be \( m:n \). According to the section formula:
\[ x = \frac{mx_1 + nx_2}{m+n} \quad \text{and} \quad y = \frac{my_1 + ny_2}{m+n} \]
Substituting the known values:
- \( x_1 = 2, y_1 = -3 \)
- \( x_2 = 5, y_2 = 6 \)
- \( x = 3, y = 0 \)

Setting up the equations:
1. For \( x \):
\[ 3 = \frac{2m + 5n}{m+n} \]
2. For \( y \):
\[ 0 = \frac{-3m + 6n}{m+n} \]

### Step 5: Solve the equations
From the second equation:
\[ -3m + 6n = 0 \implies 6n = 3m \implies 2n = m \implies m = 2n \]

Now, substitute \( m = 2n \) into the first equation:
\[ 3 = \frac{2(2n) + 5n}{2n + n} = \frac{4n + 5n}{3n} = \frac{9n}{3n} = 3 \]
This confirms that our substitution is correct.

### Step 6: Find the ratio
Since \( m = 2n \), the ratio \( m:n \) is:
\[ m:n = 2n:n = 2:1 \]

### Final Answer
The ratio in which the line segment joining (2, -3) and (5, 6) is divided by the x-axis is **2:1**.

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