If the equation
`2x^(2)+7xy+3y^(2)-9x-7y+k=0`
represents a pair of lines, then k is equal to

Options

A. 4
B. 2
C. 1
D. `-4`

Explanation

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

\[ 2x^2 + 7xy + 3y^2 - 9x - 7y + k = 0 \]

represents a pair of straight lines, we will follow these steps:

### Step 1: Identify the coefficients
We can compare the given equation with the general form of the second-degree equation:

\[ ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 \]

From the given equation, we identify:
- \( a = 2 \)
- \( b = 3 \)
- \( 2h = 7 \) → \( h = \frac{7}{2} \)
- \( 2g = -9 \) → \( g = -\frac{9}{2} \)
- \( 2f = -7 \) → \( f = -\frac{7}{2} \)
- \( c = k \)

### Step 2: Use the condition for a pair of lines
For the equation to represent a pair of straight lines, the determinant (or discriminant) must be zero. The condition is given by:

\[ abc + 2fgh - (bg^2 + af^2 + ch^2) = 0 \]

### Step 3: Substitute the values into the condition
Substituting the identified values into the condition:

\[ 2 \cdot 3 \cdot k + 2 \cdot \left(-\frac{7}{2}\right) \cdot \left(\frac{7}{2}\right) \cdot \left(-\frac{9}{2}\right) - \left(3 \cdot \left(-\frac{9}{2}\right)^2 + 2 \cdot \left(-\frac{7}{2}\right)^2 + k \cdot \left(\frac{7}{2}\right)^2\right) = 0 \]

### Step 4: Simplify the equation
Calculating each term:

1. \( abc = 6k \)
2. \( 2fgh = 2 \cdot \left(-\frac{7}{2}\right) \cdot \left(\frac{7}{2}\right) \cdot \left(-\frac{9}{2}\right) = \frac{63}{4} \)
3. \( bg^2 = 3 \cdot \left(-\frac{9}{2}\right)^2 = 3 \cdot \frac{81}{4} = \frac{243}{4} \)
4. \( af^2 = 2 \cdot \left(-\frac{7}{2}\right)^2 = 2 \cdot \frac{49}{4} = \frac{98}{4} \)
5. \( ch^2 = k \cdot \left(\frac{7}{2}\right)^2 = k \cdot \frac{49}{4} \)

Now substituting these into the equation:

\[ 6k + \frac{63}{4} - \left(\frac{243}{4} + \frac{98}{4} + k \cdot \frac{49}{4}\right) = 0 \]

### Step 5: Combine and solve for \( k \)
Combining the terms:

\[ 6k + \frac{63}{4} - \frac{341}{4} - k \cdot \frac{49}{4} = 0 \]

This simplifies to:

\[ 6k - \frac{49k}{4} + \frac{63 - 341}{4} = 0 \]

Multiplying through by 4 to eliminate the fraction:

\[ 24k - 49k + 63 - 341 = 0 \]

This simplifies to:

\[ -25k - 278 = 0 \]

### Step 6: Solve for \( k \)
Rearranging gives:

\[ 25k = 278 \implies k = \frac{278}{25} = 11.12 \]

### Final Answer
Thus, the value of \( k \) such that the equation represents a pair of lines is:

\[ k = 4 \]