What is the ratio in which x-axis divides the line joining the points `P(-4,8)` and `D(5,-4)`?

Options

A. 0.042361111111111
B. 0.17013888888889
C. 0.084027777777778
D. 0.33541666666667

Answer

To find the ratio in which the x-axis divides the line joining the points \( P(-4, 8) \) and \( D(5, -4) \), we can use the section formula. Let's go through the solution step by step.

### Step 1: Understand the Problem
We need to find the point where the x-axis intersects the line segment joining points \( P(-4, 8) \) and \( D(5, -4) \). The x-axis has a y-coordinate of 0.

### Step 2: Set Up the Section Formula
Let the point where the x-axis intersects the line segment be \( X(x, 0) \). According to the section formula, if a point divides the line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \), the coordinates of the point are given by:
\[ X = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In our case, the coordinates of points \( P \) and \( D \) are:
- \( P(-4, 8) \) where \( x_1 = -4 \) and \( y_1 = 8 \)
- \( D(5, -4) \) where \( x_2 = 5 \) and \( y_2 = -4 \)

### Step 3: Set Up the Equation for Y-Coordinate
Since the y-coordinate of point \( X \) is 0 (because it lies on the x-axis), we can set up the equation:
\[ 0 = \frac{m(-4) + n(8)}{m+n} \]

### Step 4: Cross Multiply
Cross multiplying gives us:
\[ 0 = m(-4) + n(8) \]
This simplifies to:
\[ -4m + 8n = 0 \]

### Step 5: Rearranging the Equation
Rearranging the equation gives:
\[ 4m = 8n \quad \Rightarrow \quad \frac{m}{n} = \frac{8}{4} = 2 \]
Thus, the ratio \( m:n \) is \( 2:1 \).

### Step 6: Conclusion
The ratio in which the x-axis divides the line joining the points \( P(-4, 8) \) and \( D(5, -4) \) is \( 2:1 \).