A straight line passes through the point (-5,2) and the portion of the line intercepted between the axes is divided at this point in the ratio 2:3. Find the equation of the line.

Answer

To find the equation of the line that passes through the point (-5, 2) and divides the intercepted portion between the axes in the ratio 2:3, we can follow these steps:

### Step 1: Understand the Intercept Form of the Line
The intercept form of a line is given by the equation:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept.

### Step 2: Determine the Intercepts
Let the x-intercept be \(a\) and the y-intercept be \(b\). The line intercepts the x-axis at \((a, 0)\) and the y-axis at \((0, b)\). The point \((-5, 2)\) divides the line segment between these intercepts in the ratio 2:3.

### Step 3: Use the Section Formula
According to the section formula, if a point \(P(x, y)\) divides the line segment joining \(A(x_1, y_1)\) and \(B(x_2, y_2)\) 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, \(P(-5, 2)\), \(A(0, b)\), \(B(a, 0)\), \(m = 2\), and \(n = 3\).

### Step 4: Set Up the Equations
For the x-coordinate:
\[ -5 = \frac{2 \cdot 0 + 3 \cdot a}{2 + 3} = \frac{3a}{5} \]
Multiplying both sides by 5:
\[ -25 = 3a \implies a = -\frac{25}{3} \]

For the y-coordinate:
\[ 2 = \frac{2 \cdot b + 3 \cdot 0}{2 + 3} = \frac{2b}{5} \]
Multiplying both sides by 5:
\[ 10 = 2b \implies b = 5 \]

### Step 5: Substitute the Values into the Intercept Form
Now we have \(a = -\frac{25}{3}\) and \(b = 5\). Substitute these values into the intercept form:
\[ \frac{x}{-\frac{25}{3}} + \frac{y}{5} = 1 \]
Multiplying through by \(-\frac{25}{3} \cdot 5\) to eliminate the denominators:
\[ 3x - 5y = -25 \]

### Step 6: Rearranging the Equation
Rearranging gives us the final equation of the line:
\[ 3x + 5y + 25 = 0 \]

### Final Answer
The equation of the line is:
\[ 3x + 5y + 25 = 0 \]