To find the equation of the line that is divided by the point \((-5, 4)\) in the ratio \(1:2\), we can follow these steps:
### Step 1: Understand the problem
We need to find the intercepts of the line on the x-axis and y-axis, which we will denote as \(A(a, 0)\) and \(B(0, b)\) respectively. The point \((-5, 4)\) divides the line segment \(AB\) in the ratio \(1:2\).
### Step 2: Use the section formula
According to the section formula, if a point \(P(x, y)\) divides the line segment joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of point \(P\) are given by:
\[
x = \frac{mx_2 + nx_1}{m+n}
\]
\[
y = \frac{my_2 + ny_1}{m+n}
\]
In our case, \(P(-5, 4)\), \(A(a, 0)\), \(B(0, b)\), \(m = 1\), and \(n = 2\).
### Step 3: Set up equations for x-coordinate
Using the x-coordinates:
\[
-5 = \frac{1 \cdot 0 + 2 \cdot a}{1 + 2}
\]
This simplifies to:
\[
-5 = \frac{2a}{3}
\]
Multiplying both sides by 3:
\[
-15 = 2a
\]
Thus, we find:
\[
a = -\frac{15}{2}
\]
### Step 4: Set up equations for y-coordinate
Using the y-coordinates:
\[
4 = \frac{1 \cdot b + 2 \cdot 0}{1 + 2}
\]
This simplifies to:
\[
4 = \frac{b}{3}
\]
Multiplying both sides by 3:
\[
12 = b
\]
### Step 5: Write the equation of the line
Now that we have the intercepts \(a = -\frac{15}{2}\) and \(b = 12\), we can use the intercept form of the equation of a line:
\[
\frac{x}{a} + \frac{y}{b} = 1
\]
Substituting the values of \(a\) and \(b\):
\[
\frac{x}{-\frac{15}{2}} + \frac{y}{12} = 1
\]
### Step 6: Simplify the equation
To eliminate the fractions, we can multiply through by the least common multiple of the denominators, which is \(12 \cdot 2 = 24\):
\[
24 \left(\frac{x}{-\frac{15}{2}} + \frac{y}{12}\right) = 24
\]
This gives:
\[
-32x + 2y = 24
\]
Rearranging gives us:
\[
32x - 2y + 24 = 0
\]
or, simplifying further:
\[
16x - y + 12 = 0
\]
### Final Answer
The equation of the line is:
\[
16x - y + 12 = 0
\]
