The equation of a line which passes through the point (-3, -2) and cuts of negative intercepts in the ratio of 4 : 3 is

Options

A. ` 2 x + 9y - 9 = 0 `
B. ` 3 x + 4y + 17 = 0 `
C. ` x + 2y + 12 = 0`
D. ` 2 x + 3y + 11 = 0 `

Answer

To find the equation of a line that passes through the point (-3, -2) and cuts negative intercepts in the ratio of 4:3, we can follow these steps:

### Step 1: Define the Intercepts
Let the x-intercept be \( a \) and the y-intercept be \( b \). According to the problem, the ratio of the intercepts is given as:
\[ \frac{a}{b} = \frac{4}{3} \]
This implies:
\[ a = -\frac{4}{3}b \]
(Note: The intercepts are negative since the line cuts the axes in the negative quadrant.)

### Step 2: Write the Equation of the Line
The equation of a line in terms of its intercepts is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
Substituting \( a = -\frac{4}{3}b \) into the equation, we have:
\[ \frac{x}{-\frac{4}{3}b} + \frac{y}{b} = 1 \]
Multiplying through by \( -\frac{4}{3}b \) to eliminate the denominators:
\[ 3x + 4y = -\frac{4}{3}b \]

### Step 3: Substitute the Point (-3, -2)
Since the line passes through the point (-3, -2), we substitute \( x = -3 \) and \( y = -2 \) into the equation:
\[ 3(-3) + 4(-2) = -\frac{4}{3}b \]
Calculating the left side:
\[ -9 - 8 = -\frac{4}{3}b \]
\[ -17 = -\frac{4}{3}b \]

### Step 4: Solve for \( b \)
To find \( b \), we can multiply both sides by -1:
\[ 17 = \frac{4}{3}b \]
Now, multiply both sides by \( \frac{3}{4} \):
\[ b = \frac{3 \times 17}{4} = \frac{51}{4} \]

### Step 5: Find \( a \)
Using the value of \( b \) to find \( a \):
\[ a = -\frac{4}{3}b = -\frac{4}{3} \times \frac{51}{4} = -\frac{51}{3} \]

### Step 6: Write the Final Equation
Now substituting \( a \) and \( b \) back into the equation of the line:
\[ \frac{x}{-\frac{51}{3}} + \frac{y}{\frac{51}{4}} = 1 \]
Multiplying through by \( -12 \) (the least common multiple of 3 and 4) to clear the denominators:
\[ -12 \cdot \frac{x}{-\frac{51}{3}} + 12 \cdot \frac{y}{\frac{51}{4}} = -12 \]
This simplifies to:
\[ 4x + 3y = -12 \]
Rearranging gives:
\[ 4x + 3y + 12 = 0 \]

### Final Answer
The equation of the line is:
\[ 4x + 3y + 12 = 0 \]