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

Options

A. 8x + 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 segment that is divided by the point (-5, 4) in the ratio 1:2 between the coordinate axes, we can follow these steps:

### Step 1: Identify the Points on the Axes
Let the points where the line intersects the x-axis and y-axis be A and B, respectively.
- Point A (x-intercept) can be represented as (a, 0).
- Point B (y-intercept) can be represented as (0, b).

### Step 2: Use the Section Formula
According to the section formula, if a point P(x, y) divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

In our case, the point P is (-5, 4), m = 1, n = 2, and the coordinates of A and B are (a, 0) and (0, b).

### Step 3: Set Up the Equations
Using the section formula for x-coordinate:
\[ -5 = \frac{1 \cdot 0 + 2 \cdot a}{1 + 2} = \frac{2a}{3} \]
From this, we can solve for a:
\[ -5 \cdot 3 = 2a \implies -15 = 2a \implies a = -\frac{15}{2} \]

Using the section formula for y-coordinate:
\[ 4 = \frac{1 \cdot b + 2 \cdot 0}{1 + 2} = \frac{b}{3} \]
From this, we can solve for b:
\[ 4 \cdot 3 = b \implies b = 12 \]

### Step 4: Identify Coordinates of A and B
Now we have:
- Point A (x-intercept) = \((-15/2, 0)\)
- Point B (y-intercept) = \((0, 12)\)

### Step 5: Find the Equation of the Line
Using the two points A and B, we can find the slope (m) of the line:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{12 - 0}{0 - (-15/2)} = \frac{12}{15/2} = \frac{12 \cdot 2}{15} = \frac{24}{15} = \frac{8}{5} \]

Now, using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Using point B (0, 12):
\[ y - 12 = \frac{8}{5}(x - 0) \implies y - 12 = \frac{8}{5}x \]
Rearranging gives:
\[ y = \frac{8}{5}x + 12 \]

### Step 6: Convert to Standard Form
To convert this to standard form (Ax + By + C = 0):
\[ -8x + 5y - 60 = 0 \]
or
\[ 8x - 5y + 60 = 0 \]

### Final Answer
The equation of the line is:
\[ 8x - 5y + 60 = 0 \]