For any two complex numbers `z_(1) "and" z_(2)`, `abs(z_(1)-z_(2)) ge {{:(abs(z_(1))-abs(z_(2))),(abs(z_(2))-abs(z_(1))):}` and equality holds iff origin `z_(1) " and " z_(2)` are collinear and `z_(1),z_(2)` lie on the same side of the origin . If `abs(z-(2)/(z))=4` and sum of greatest and least values of `abs(z)` is `lambda`, then `lambda^(2)`, is

Options

A. 12
B. 18
C. 24
D. 30

Answer

To solve the given problem step by step, we start with the equation provided:

### Step 1: Start with the given equation
We have:
\[ \frac{|z - 2|}{|z|} = 4 \]

### Step 2: Rewrite the equation
From the equation, we can express it as:
\[ |z - 2| = 4 |z| \]

### Step 3: Apply the triangle inequality
Using the triangle inequality, we know:
\[ |z - 2| \geq ||z| - 2| \]
Thus, we can write:
\[ 4 |z| \geq ||z| - 2| \]

### Step 4: Consider two cases for the absolute value
This gives us two cases to consider:
1. \(4 |z| \geq |z| - 2\)
2. \(4 |z| \geq -(|z| - 2)\)

### Step 5: Solve the first case
For the first case:
\[ 4 |z| \geq |z| - 2 \]
Rearranging gives:
\[ 3 |z| \geq -2 \quad \Rightarrow \quad |z| \geq -\frac{2}{3} \]
This is always true since \(|z|\) is non-negative.

### Step 6: Solve the second case
For the second case:
\[ 4 |z| \geq -|z| + 2 \]
Rearranging gives:
\[ 5 |z| \geq 2 \quad \Rightarrow \quad |z| \geq \frac{2}{5} \]

### Step 7: Combine the inequalities
Now we have:
\[ |z| \geq \frac{2}{5} \]
and from our earlier manipulation, we also need to consider the upper bound.

### Step 8: Find the upper bound
From the equation \( |z - 2| = 4 |z| \), we can also find the maximum value of \(|z|\). We rewrite it as:
\[ |z| = \frac{|z - 2|}{4} \]
Substituting \(|z - 2|\) gives us the quadratic inequality:
\[ |z|^2 - 4|z| - 2 \leq 0 \]

### Step 9: Solve the quadratic inequality
Using the quadratic formula:
\[ |z| = \frac{4 \pm \sqrt{16 + 8}}{2} = \frac{4 \pm \sqrt{24}}{2} = 2 \pm \sqrt{6} \]
Thus, the roots are:
\[ |z| = 2 + \sqrt{6} \quad \text{and} \quad |z| = 2 - \sqrt{6} \]

### Step 10: Determine the range of \(|z|\)
The valid range for \(|z|\) is:
\[ 2 - \sqrt{6} \leq |z| \leq 2 + \sqrt{6} \]

### Step 11: Calculate the sum of greatest and least values
The least value is \(2 - \sqrt{6}\) and the greatest value is \(2 + \sqrt{6}\). Therefore, the sum is:
\[ \lambda = (2 - \sqrt{6}) + (2 + \sqrt{6}) = 4 \]

### Step 12: Calculate \(\lambda^2\)
Finally, we find:
\[ \lambda^2 = 4^2 = 16 \]

### Final Answer
Thus, the answer is:
\[ \lambda^2 = 16 \]

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