A line passes through the point (-2, 6) and portion of the line intercepted between the axes is divided at this point in the ratio 3:2 . Find the equation of the line.

Options

A. `2x-y=10`
B. `2x-y+10=0`
C. `x-2y=10`
D. `x+2y=10`

Answer

To find the equation of the line that passes through the point (-2, 6) and divides the intercepted portion between the axes in the ratio 3:2, we can follow these steps:

### Step 1: Understand the Intercepts
Let the x-intercept of the line be \( a \) (where the line meets the x-axis) and the y-intercept be \( b \) (where the line meets the y-axis). The coordinates of these intercepts are \( (a, 0) \) and \( (0, b) \).

### Step 2: Apply the Section Formula
Since the point (-2, 6) divides the line segment between the intercepts in the ratio 3:2, we can use the section formula. The section formula states that 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:

\[ x = \frac{m x_2 + n x_1}{m+n} \quad \text{and} \quad y = \frac{m y_2 + n y_1}{m+n} \]

In our case, \( m = 3 \), \( n = 2 \), \( A(0, b) \), and \( B(a, 0) \).

### Step 3: Set Up the Equations
Using the section formula for the x-coordinate:

\[ -2 = \frac{3(0) + 2(a)}{3 + 2} = \frac{2a}{5} \]

Multiplying both sides by 5:

\[ -10 = 2a \implies a = -5 \]

Now, for the y-coordinate:

\[ 6 = \frac{3(b) + 2(0)}{3 + 2} = \frac{3b}{5} \]

Multiplying both sides by 5:

\[ 30 = 3b \implies b = 10 \]

### Step 4: Write the Equation of the Line
Now that we have the intercepts \( a = -5 \) and \( b = 10 \), we can write the equation of the line in intercept form:

\[ \frac{x}{a} + \frac{y}{b} = 1 \implies \frac{x}{-5} + \frac{y}{10} = 1 \]

Multiplying through by -10 to eliminate the fractions:

\[ -2x + y = 10 \]

Rearranging gives us:

\[ 2x - y + 10 = 0 \]

### Final Equation
Thus, the equation of the line is:

\[ 2x - y + 10 = 0 \]