Find the complex number z with maximum and minimum possible values of `|z|` satisfying
(a) `|z + (1)/(z) | =1`.
(b) `|z+ (4)/(z)| =3`.

Answer

To find the complex number \( z \) with maximum and minimum possible values of \( |z| \) satisfying the given conditions, we will solve each part step by step.

### Part (a): \( |z + \frac{1}{z}| = 1 \)

1. **Start with the given equation:**
\[ |z + \frac{1}{z}| = 1 \]

2. **Use the triangle inequality:**
\[ |z + \frac{1}{z}| \leq |z| + \left|\frac{1}{z}\right| = |z| + \frac{1}{|z|} \]
Thus, we have:
\[ 1 \leq |z| + \frac{1}{|z|} \]

3. **Let \( r = |z| \):**
\[ 1 \leq r + \frac{1}{r} \]

4. **Rearranging gives:**
\[ r + \frac{1}{r} \geq 1 \]

5. **Multiply through by \( r \) (assuming \( r > 0 \)):**
\[ r^2 - r + 1 \geq 0 \]

6. **Find the roots of the equation \( r^2 - r + 1 = 0 \):**
\[ D = (-1)^2 - 4 \cdot 1 \cdot 1 = 1 - 4 = -3 \]
Since the discriminant is negative, the quadratic has no real roots, meaning \( r^2 - r + 1 \) is always positive.

7. **Now, consider the equality case:**
\[ |z + \frac{1}{z}| = |z| + \left|\frac{1}{z}\right| \implies z \text{ and } \frac{1}{z} \text{ are in the same direction.} \]

8. **Set \( z = re^{i\theta} \):**
\[ |re^{i\theta} + \frac{1}{re^{i\theta}}| = 1 \]
This simplifies to:
\[ |r + \frac{1}{r}| = 1 \]

9. **Solve the equation \( r + \frac{1}{r} = 1 \):**
\[ r^2 - r + 1 = 0 \text{ (already shown has no real roots)} \]

10. **Now consider the critical points:**
\[ r + \frac{1}{r} = 1 \implies r^2 - r + 1 = 0 \text{ has no solutions.} \]
Thus, we analyze the inequality:
\[ r + \frac{1}{r} \geq 2 \implies r \geq 1 \]

11. **Minimum and maximum values:**
- Minimum value of \( |z| = 1 - \frac{\sqrt{5}}{2} \)
- Maximum value of \( |z| = 1 + \frac{\sqrt{5}}{2} \)

### Part (b): \( |z + \frac{4}{z}| = 3 \)

1. **Start with the given equation:**
\[ |z + \frac{4}{z}| = 3 \]

2. **Use the triangle inequality:**
\[ |z + \frac{4}{z}| \leq |z| + \left|\frac{4}{z}\right| = |z| + \frac{4}{|z|} \]
Thus, we have:
\[ 3 \leq |z| + \frac{4}{|z|} \]

3. **Let \( r = |z| \):**
\[ 3 \leq r + \frac{4}{r} \]

4. **Rearranging gives:**
\[ r + \frac{4}{r} \geq 3 \]

5. **Multiply through by \( r \) (assuming \( r > 0 \)):**
\[ r^2 - 3r + 4 \geq 0 \]

6. **Find the roots of the equation \( r^2 - 3r + 4 = 0 \):**
\[ D = (-3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7 \]
Since the discriminant is negative, the quadratic has no real roots, meaning \( r^2 - 3r + 4 \) is always positive.

7. **Now consider the equality case:**
\[ |z + \frac{4}{z}| = |z| + \left|\frac{4}{z}\right| \implies z \text{ and } \frac{4}{z} \text{ are in the same direction.} \]

8. **Set \( z = re^{i\theta} \):**
\[ |re^{i\theta} + \frac{4}{re^{i\theta}}| = 3 \]
This simplifies to:
\[ |r + \frac{4}{r}| = 3 \]

9. **Solve the equation \( r + \frac{4}{r} = 3 \):**
\[ r^2 - 3r + 4 = 0 \text{ (already shown has no real roots)} \]

10. **Now consider the critical points:**
\[ r + \frac{4}{r} = 3 \implies r^2 - 3r + 4 = 0 \text{ has no solutions.} \]
Thus, we analyze the inequality:
\[ r + \frac{4}{r} \geq 3 \implies r \geq 1 \]

11. **Minimum and maximum values:**
- Minimum value of \( |z| = 1 \)
- Maximum value of \( |z| = 4 \)

### Summary of Results:
- For part (a):
- Minimum value of \( |z| = 1 - \frac{\sqrt{5}}{2} \)
- Maximum value of \( |z| = 1 + \frac{\sqrt{5}}{2} \)

- For part (b):
- Minimum value of \( |z| = 1 \)
- Maximum value of \( |z| = 4 \)