If `|z| ge 5`, then least value of `|z - (1)/(z)|` is

Options

A. 5
B. `24//5`
C. 8
D. `8//3`

Answer

To find the least value of \( |z - \frac{1}{z}| \) given that \( |z| \geq 5 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression we need to minimize:
\[ |z - \frac{1}{z}| \]

### Step 2: Use the property of moduli
Using the property of moduli, we can express this as:
\[ |z - \frac{1}{z}| = |z| \cdot |1 - \frac{1}{z^2}| \]
This is because \( |a - b| = |a| \cdot |1 - \frac{b}{a}| \).

### Step 3: Substitute the modulus of z
Since we know that \( |z| \geq 5 \), we can denote \( |z| = r \) where \( r \geq 5 \). Thus, we rewrite the expression:
\[ |z - \frac{1}{z}| = r \cdot |1 - \frac{1}{r^2}| \]

### Step 4: Simplify the expression
Now, we need to simplify \( |1 - \frac{1}{r^2}| \):
\[ |1 - \frac{1}{r^2}| = | \frac{r^2 - 1}{r^2} | = \frac{r^2 - 1}{r^2} \]
This is valid since \( r^2 \) is always positive for \( r \geq 5 \).

### Step 5: Combine the expressions
Now we can combine the expressions:
\[ |z - \frac{1}{z}| = r \cdot \frac{r^2 - 1}{r^2} = \frac{r(r^2 - 1)}{r^2} = \frac{r^3 - r}{r^2} = r - \frac{1}{r} \]

### Step 6: Minimize the expression
To find the minimum value of \( r - \frac{1}{r} \) under the constraint \( r \geq 5 \), we can analyze the function \( f(r) = r - \frac{1}{r} \).

### Step 7: Calculate the value at the boundary
Since \( r \) is minimized at \( r = 5 \):
\[ f(5) = 5 - \frac{1}{5} = 5 - 0.2 = 4.8 \]

### Conclusion
Thus, the least value of \( |z - \frac{1}{z}| \) when \( |z| \geq 5 \) is:
\[ \boxed{4.8} \]

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