The maximum value of |z| where z satisfies the condition` |z+(2/z)|=2` is

Options

A. `sqrt3-1`
B. `sqrt3+1`
C. `sqrt3`
D. `sqrt2 +sqrt3`

Answer

To solve the problem of finding the maximum value of |z| where z satisfies the condition |z + (2/z)| = 2, we can follow these steps:

### Step 1: Rewrite the given condition
We start with the equation:
\[ |z + \frac{2}{z}| = 2 \]

### Step 2: Introduce a new variable
Let \( z = r e^{i\theta} \), where \( r = |z| \) (the modulus of z) and \( \theta \) is the argument of z. Then, we can express \( \frac{2}{z} \) as:
\[ \frac{2}{z} = \frac{2}{r e^{i\theta}} = \frac{2}{r} e^{-i\theta} \]

### Step 3: Substitute into the modulus equation
Now we substitute this into the modulus equation:
\[ |r e^{i\theta} + \frac{2}{r} e^{-i\theta}| = 2 \]

### Step 4: Simplify the expression
Using the property of modulus, we can write:
\[ |r e^{i\theta} + \frac{2}{r} e^{-i\theta}| = |r + \frac{2}{r} e^{-2i\theta}| \]
This simplifies to:
\[ \sqrt{(r + \frac{2}{r} \cos(-\theta))^2 + (\frac{2}{r} \sin(-\theta))^2} = 2 \]

### Step 5: Square both sides
Squaring both sides gives:
\[ (r + \frac{2}{r} \cos \theta)^2 + (\frac{2}{r} \sin \theta)^2 = 4 \]

### Step 6: Expand and simplify
Expanding the left side:
\[ r^2 + 2 \cdot r \cdot \frac{2}{r} \cos \theta + \left(\frac{2}{r}\right)^2 \cos^2 \theta + \left(\frac{2}{r}\right)^2 \sin^2 \theta = 4 \]
This simplifies to:
\[ r^2 + 4 \cos \theta + \frac{4}{r^2} = 4 \]

### Step 7: Rearrange the equation
Rearranging gives:
\[ r^2 + \frac{4}{r^2} + 4 \cos \theta - 4 = 0 \]

### Step 8: Use the Cauchy-Schwarz inequality
To maximize |z|, we can apply the Cauchy-Schwarz inequality:
\[ |z|^2 + |2/z|^2 \geq |z + 2/z|^2 \]
This leads to:
\[ |z|^2 + \frac{4}{|z|^2} \geq 4 \]

### Step 9: Let \( x = |z|^2 \)
Let \( x = |z|^2 \). Then we have:
\[ x + \frac{4}{x} \geq 4 \]

### Step 10: Solve the inequality
Multiplying through by x (assuming x > 0) gives:
\[ x^2 - 4x + 4 \geq 0 \]
This factors to:
\[ (x - 2)^2 \geq 0 \]
This is always true, with equality when \( x = 2 \).

### Step 11: Find the maximum value of |z|
Thus, the maximum value of |z| is:
\[ |z| = \sqrt{x} = \sqrt{2} \]

### Step 12: Conclusion
The maximum value of |z| is:
\[ \sqrt{2} \]