If `2x^(2)+5xy+2y^(2)-11x-7y+K=0` is the pair of asymptotes of the hyperbola `2x^(2)+5xy+2y^(2)-11x-7y-4=0` then K=

Options

A. 3
B. 4
C. 5
D. 6

Answer

To find the value of \( K \) such that the equation \( 2x^2 + 5xy + 2y^2 - 11x - 7y + K = 0 \) represents the pair of asymptotes of the hyperbola given by \( 2x^2 + 5xy + 2y^2 - 11x - 7y - 4 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients of the hyperbola equation
The hyperbola equation is:
\[ 2x^2 + 5xy + 2y^2 - 11x - 7y - 4 = 0 \]
From this, we can identify the coefficients:
- \( a = 2 \)
- \( b = 2 \)
- \( c = -4 \)
- \( g = -\frac{11}{2} \)
- \( f = -\frac{7}{2} \)
- \( h = \frac{5}{2} \)

### Step 2: Compare the asymptote equation
The asymptote equation is:
\[ 2x^2 + 5xy + 2y^2 - 11x - 7y + K = 0 \]
From this, we can identify the coefficients:
- \( a = 2 \)
- \( b = 2 \)
- \( c = K \)
- \( g = -\frac{11}{2} \)
- \( f = -\frac{7}{2} \)
- \( h = \frac{5}{2} \)

### Step 3: Use the condition for asymptotes
For the equation to represent the pair of asymptotes of the hyperbola, the following condition must hold:
\[ abc + 2fgh - af^2 - bg^2 - ch^2 = 0 \]
Substituting the values we have:
- \( a = 2 \)
- \( b = 2 \)
- \( c = K \)
- \( f = -\frac{7}{2} \)
- \( g = -\frac{11}{2} \)
- \( h = \frac{5}{2} \)

### Step 4: Substitute and simplify
Substituting these values into the condition:
\[ 2 \cdot 2 \cdot K + 2 \left(-\frac{7}{2}\right) \left(-\frac{11}{2}\right) \left(\frac{5}{2}\right) - 2 \left(-\frac{7}{2}\right)^2 - 2 \left(-\frac{11}{2}\right)^2 - K \left(\frac{5}{2}\right)^2 = 0 \]

Calculating each term:
1. \( 2 \cdot 2 \cdot K = 4K \)
2. \( 2 \left(-\frac{7}{2}\right) \left(-\frac{11}{2}\right) \left(\frac{5}{2}\right) = 2 \cdot \frac{77}{4} \cdot \frac{5}{2} = \frac{385}{4} \)
3. \( -2 \left(-\frac{7}{2}\right)^2 = -2 \cdot \frac{49}{4} = -\frac{98}{4} \)
4. \( -2 \left(-\frac{11}{2}\right)^2 = -2 \cdot \frac{121}{4} = -\frac{242}{4} \)
5. \( -K \left(\frac{5}{2}\right)^2 = -K \cdot \frac{25}{4} \)

Putting it all together:
\[ 4K + \frac{385}{4} - \frac{98}{4} - \frac{242}{4} - \frac{25K}{4} = 0 \]
Combine the constants:
\[ 4K - \frac{25K}{4} + \frac{385 - 98 - 242}{4} = 0 \]
This simplifies to:
\[ 4K - \frac{25K}{4} + \frac{45}{4} = 0 \]
Multiply through by 4 to eliminate the fraction:
\[ 16K - 25K + 45 = 0 \]
Combine like terms:
\[ -9K + 45 = 0 \]
Solving for \( K \):
\[ 9K = 45 \implies K = 5 \]

### Final Answer
Thus, the value of \( K \) is \( \boxed{5} \).