If k is the largest possible real number such that ` p^(4) + q^(4) = (p^(2) + kpq + q^(2)) (p^(2) - kpq + q^(2))` then the value of k is

Options

A. A)1
B. B)`- sqrt(2)`
C. C)2
D. D)`sqrt(2)`

Answer

To solve the equation \( p^4 + q^4 = (p^2 + kpq + q^2)(p^2 - kpq + q^2) \) and find the largest possible real number \( k \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ p^4 + q^4 = (p^2 + kpq + q^2)(p^2 - kpq + q^2) \]

### Step 2: Expand the right-hand side
Using the difference of squares, we can expand the right-hand side:
\[ (p^2 + kpq + q^2)(p^2 - kpq + q^2) = (p^2 + q^2)^2 - (kpq)^2 \]
This simplifies to:
\[ p^4 + 2p^2q^2 + q^4 - k^2p^2q^2 \]
Combining like terms gives:
\[ p^4 + q^4 + (2 - k^2)p^2q^2 \]

### Step 3: Set the equation equal
Now we set the two sides equal:
\[ p^4 + q^4 = p^4 + q^4 + (2 - k^2)p^2q^2 \]

### Step 4: Simplify the equation
Subtract \( p^4 + q^4 \) from both sides:
\[ 0 = (2 - k^2)p^2q^2 \]

### Step 5: Analyze the equation
For this equation to hold for all \( p \) and \( q \), the coefficient must be zero:
\[ 2 - k^2 = 0 \]

### Step 6: Solve for \( k \)
Solving for \( k^2 \) gives:
\[ k^2 = 2 \]
Thus, taking the square root gives:
\[ k = \sqrt{2} \quad \text{or} \quad k = -\sqrt{2} \]

### Step 7: Determine the largest value
Since we are looking for the largest possible real number \( k \), we choose:
\[ k = \sqrt{2} \]

### Final Answer
The value of \( k \) is:
\[ \boxed{\sqrt{2}} \]