Let `abs(a)=abs(b)=2" and "p=a+b, q=a-b`. If `abs(p times q)=2(k-(a.b)^(2))^(1//2)` then

Options

A. k = 16
B. k = 8
C. k = 4
D. k = 1

Answer

To solve the problem, we need to find the value of \( k \) given the conditions. Let's break it down step by step.

### Step 1: Define the vectors and their magnitudes
We are given that \( |a| = |b| = 2 \). This means:
\[ |a|^2 = 4 \quad \text{and} \quad |b|^2 = 4 \]

### Step 2: Define the vectors \( p \) and \( q \)
We define:
\[ p = a + b \quad \text{and} \quad q = a - b \]

### Step 3: Calculate the cross product \( p \times q \)
We need to find \( p \times q \):
\[ p \times q = (a + b) \times (a - b) \]

Using the distributive property of the cross product:
\[ p \times q = a \times a - a \times b + b \times a - b \times b \]

### Step 4: Simplify the cross product
Since the cross product of any vector with itself is zero:
\[ a \times a = 0 \quad \text{and} \quad b \times b = 0 \]
Thus, we have:
\[ p \times q = -a \times b + b \times a = -a \times b - a \times b = -2(a \times b) \]

### Step 5: Find the magnitude of \( p \times q \)
Now, we need the magnitude:
\[ |p \times q| = |-2(a \times b)| = 2 |a \times b| \]

### Step 6: Calculate \( |a \times b| \)
The magnitude of the cross product can be expressed as:
\[ |a \times b| = |a||b| \sin \theta \]
where \( \theta \) is the angle between vectors \( a \) and \( b \). Given \( |a| = |b| = 2 \):
\[ |a \times b| = 2 \cdot 2 \cdot \sin \theta = 4 \sin \theta \]

### Step 7: Substitute back into the magnitude expression
Thus, we have:
\[ |p \times q| = 2 |a \times b| = 2 \cdot 4 \sin \theta = 8 \sin \theta \]

### Step 8: Relate to the given expression
According to the problem, we have:
\[ |p \times q| = 2 \sqrt{k - (a \cdot b)^2} \]

### Step 9: Find \( a \cdot b \)
Using the dot product:
\[ a \cdot b = |a||b| \cos \theta = 2 \cdot 2 \cdot \cos \theta = 4 \cos \theta \]
Thus,
\[ (a \cdot b)^2 = (4 \cos \theta)^2 = 16 \cos^2 \theta \]

### Step 10: Set the expressions equal
Now we can equate the two expressions:
\[ 8 \sin \theta = 2 \sqrt{k - 16 \cos^2 \theta} \]

### Step 11: Simplify the equation
Dividing both sides by 2:
\[ 4 \sin \theta = \sqrt{k - 16 \cos^2 \theta} \]

### Step 12: Square both sides
Squaring both sides gives:
\[ 16 \sin^2 \theta = k - 16 \cos^2 \theta \]

### Step 13: Rearranging the equation
Rearranging gives:
\[ k = 16 \sin^2 \theta + 16 \cos^2 \theta \]
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ k = 16(1) = 16 \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{16} \]