If `z=x+iy and x^2+y^2=16 ,` then the range of `||x|-|y||` is `[0,4]` b. `[0,2]` c. `[2,4]` d. none of these

Options

A. `[0,4]`
B. `[0,2]`
C. `[2,4]`
D. none of these

Explanation

To find the range of \( ||x| - |y|| \) given that \( z = x + iy \) and \( x^2 + y^2 = 16 \), we can follow these steps:

### Step 1: Understand the given equation
The equation \( x^2 + y^2 = 16 \) represents a circle with a radius of 4 centered at the origin in the xy-plane. This means that the values of \( x \) and \( y \) can be expressed in terms of trigonometric functions.

### Step 2: Parametrize \( x \) and \( y \)
We can express \( x \) and \( y \) in terms of a parameter \( \theta \):
\[ x = 4 \cos \theta, \quad y = 4 \sin \theta \]

### Step 3: Calculate \( ||x| - |y|| \)
Now we need to find \( ||x| - |y|| \):
\[ ||x| - |y|| = ||4 \cos \theta| - |4 \sin \theta|| = 4 ||\cos \theta - \sin \theta|| \]

### Step 4: Simplify the expression
We can simplify this to:
\[ ||x| - |y|| = 4 |\cos \theta - \sin \theta| \]

### Step 5: Find the maximum and minimum values of \( |\cos \theta - \sin \theta| \)
To find the range of \( |\cos \theta - \sin \theta| \), we can use the identity:
\[ |\cos \theta - \sin \theta| = \sqrt{(\cos \theta - \sin \theta)^2} \]
Expanding this gives:
\[ (\cos \theta - \sin \theta)^2 = \cos^2 \theta + \sin^2 \theta - 2 \cos \theta \sin \theta = 1 - \sin 2\theta \]
Thus:
\[ |\cos \theta - \sin \theta| = \sqrt{1 - \sin 2\theta} \]

### Step 6: Determine the range of \( \sin 2\theta \)
The function \( \sin 2\theta \) varies between -1 and 1. Therefore:
- The maximum value of \( 1 - \sin 2\theta \) occurs when \( \sin 2\theta = -1 \):
\[ 1 - (-1) = 2 \quad \Rightarrow \quad \sqrt{2} \]
- The minimum value occurs when \( \sin 2\theta = 1 \):
\[ 1 - 1 = 0 \quad \Rightarrow \quad \sqrt{0} = 0 \]

### Step 7: Calculate the range of \( ||x| - |y|| \)
Thus, the range of \( |\cos \theta - \sin \theta| \) is from 0 to \( \sqrt{2} \). Therefore, the range of \( ||x| - |y|| \) becomes:
\[ 4 \cdot [0, \sqrt{2}] = [0, 4\sqrt{2}] \]

### Step 8: Final result
Since \( 4\sqrt{2} \) is approximately \( 5.66 \), we conclude that the range of \( ||x| - |y|| \) is:
\[ [0, 4] \]

### Conclusion
The correct answer is \( [0, 4] \).