If the equation `2x^(2)+4xy-2y^(2)+4x+8y+k=0` represents a pair of line, then k=

Options

A. `1`
B. `-1`
C. `8`
D. `-8`

Answer

To determine the value of \( k \) such that the equation

\[ 2x^2 + 4xy - 2y^2 + 4x + 8y + k = 0 \]

represents a pair of straight lines, we will use the condition for a conic section to represent a pair of lines, which is given by the determinant:

\[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0 \]

where the general form of the conic is \( ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \).

### Step 1: Identify coefficients
From the given equation, we can identify the coefficients:
- \( a = 2 \)
- \( b = -2 \)
- \( h = \frac{4}{2} = 2 \)
- \( g = 2 \)
- \( f = \frac{8}{2} = 4 \)
- \( c = k \)

### Step 2: Set up the determinant
Now we set up the determinant:

\[ \begin{vmatrix} 2 & 2 & 2 \\ 2 & -2 & 4 \\ 2 & 4 & k \end{vmatrix} = 0 \]

### Step 3: Calculate the determinant
We will calculate the determinant using the formula for a 3x3 matrix:

\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Substituting the values:

\[ D = 2((-2)k - 4 \cdot 2) - 2(2k - 4 \cdot 2) + 2(2 \cdot 4 - (-2) \cdot 2) \]

Calculating each part:

1. First term: \( 2((-2)k - 8) = 2(-2k - 8) = -4k - 16 \)
2. Second term: \( -2(2k - 8) = -4k + 16 \)
3. Third term: \( 2(8 + 4) = 2 \cdot 12 = 24 \)

Combining these:

\[ -4k - 16 - 4k + 16 + 24 = 0 \]

### Step 4: Simplify the equation
This simplifies to:

\[ -8k + 24 = 0 \]

### Step 5: Solve for \( k \)
Now, we can solve for \( k \):

\[ -8k = -24 \implies k = \frac{-24}{-8} = 3 \]

### Final Answer
Thus, the value of \( k \) is:

\[ \boxed{3} \]