If (-5, 4) divides the line segment between the coordinate axes in the ratio 1: 2, then what is its equation?

Options

A. 8 x + 5y + 20 = 0
B. 5x +8y - 7 = 0
C. 8x - 5y + 60 = 0
D. 5x - 8y + 57 = 0

Answer

To find the equation of the line that divides the line segment between the coordinate axes at the point (-5, 4) in the ratio 1:2, we can follow these steps:

### Step 1: Identify the coordinates of the points on the axes
Let the point where the line intersects the y-axis be \( P(0, b) \) and the point where it intersects the x-axis be \( Q(a, 0) \).

### Step 2: Use the section formula
The point (-5, 4) divides the line segment \( PQ \) in the ratio 1:2. According to the section formula, the coordinates of the point dividing the line segment in the ratio \( m:n \) are given by:

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

Here, \( m = 1 \), \( n = 2 \), \( P(0, b) \) corresponds to \( (x_1, y_1) \) and \( Q(a, 0) \) corresponds to \( (x_2, y_2) \).

### Step 3: Set up the equations
Using the section formula, we can set up the following equations for the x-coordinate and y-coordinate:

For the x-coordinate:
\[ -5 = \frac{1 \cdot a + 2 \cdot 0}{1 + 2} = \frac{a}{3} \]

For the y-coordinate:
\[ 4 = \frac{1 \cdot 0 + 2 \cdot b}{1 + 2} = \frac{2b}{3} \]

### Step 4: Solve for \( a \) and \( b \)
From the x-coordinate equation:
\[ -5 = \frac{a}{3} \implies a = -15 \]

From the y-coordinate equation:
\[ 4 = \frac{2b}{3} \implies 2b = 12 \implies b = 6 \]

### Step 5: Identify the coordinates of points \( P \) and \( Q \)
Now we have the coordinates:
- Point \( P(0, 6) \)
- Point \( Q(-15, 0) \)

### Step 6: Find the equation of the line
The slope \( m \) of the line passing through points \( P(0, 6) \) and \( Q(-15, 0) \) can be calculated as follows:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 6}{-15 - 0} = \frac{-6}{-15} = \frac{2}{5} \]

Now, using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using point \( P(0, 6) \):
\[ y - 6 = \frac{2}{5}(x - 0) \]
\[ y - 6 = \frac{2}{5}x \]
\[ y = \frac{2}{5}x + 6 \]

### Final Equation
Thus, the equation of the line is:
\[ y = \frac{2}{5}x + 6 \]