If `2x + y = 0` is one of the lines represented by `3x^(2) + kxy + 2y^(2) = 0`. Then the value of k is

Options

A. `(1)/(2)`
B. `(11)/(2)`
C. `(5)/(2)`
D. `-(11)/(2)`

Explanation

To find the value of \( k \) such that the line \( 2x + y = 0 \) is one of the lines represented by the equation \( 3x^2 + kxy + 2y^2 = 0 \), we can follow these steps:

### Step 1: Identify the slope of the line
The given line is \( 2x + y = 0 \). We can rewrite this in slope-intercept form \( y = mx + c \):
\[ y = -2x \]
Thus, the slope \( m \) of the line is \( -2 \).

**Hint:** Convert the line equation into slope-intercept form to easily identify the slope.

### Step 2: Substitute \( y \) in terms of \( x \)
We know that the equation \( 3x^2 + kxy + 2y^2 = 0 \) represents a pair of straight lines. To analyze this, we can substitute \( y = mx \) into the equation. Since we have \( m = -2 \), we can write:
\[ y = -2x \]

**Hint:** Use the relationship \( y = mx \) to substitute \( y \) in the quadratic equation.

### Step 3: Substitute into the quadratic equation
Substituting \( y = -2x \) into the equation \( 3x^2 + kxy + 2y^2 = 0 \):
\[ 3x^2 + kx(-2x) + 2(-2x)^2 = 0 \]
This simplifies to:
\[ 3x^2 - 2kx^2 + 8x^2 = 0 \]
Combining like terms gives:
\[ (3 - 2k + 8)x^2 = 0 \]
Thus, we have:
\[ (11 - 2k)x^2 = 0 \]

**Hint:** Combine the terms carefully to form a single quadratic equation in \( x \).

### Step 4: Set the coefficient to zero
For the equation to represent a line, the coefficient of \( x^2 \) must be zero:
\[ 11 - 2k = 0 \]

**Hint:** Set the coefficient of \( x^2 \) to zero to find the necessary condition for \( k \).

### Step 5: Solve for \( k \)
Now, solving for \( k \):
\[ 2k = 11 \implies k = \frac{11}{2} \]

**Hint:** Isolate \( k \) by performing algebraic operations.

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{\frac{11}{2}} \]