The maximum distance from the origin of coordinates to the point z satisfying the equation `|z+1/z|=a` is

Options

A. `1/2(sqrt(a^(2)+1)+a)`
B. `1/2(sqrt(a^(2)+2)+a)`
C. `1/2(sqrt(a^(2)-4)+a)`
D. `1/2(sqrt(a^(2)+1)-a)`

Answer

To solve the problem of finding the maximum distance from the origin to the point \( z \) satisfying the equation \( |z + \frac{1}{z}| = a \), we can follow these steps:

### Step 1: Understanding the Equation
The equation \( |z + \frac{1}{z}| = a \) can be interpreted in terms of complex numbers. Here, \( z \) can be expressed in polar form as \( z = re^{i\theta} \), where \( r = |z| \) is the modulus (distance from the origin) and \( \theta \) is the argument.

### Step 2: Rewrite the Expression
We rewrite the expression \( |z + \frac{1}{z}| \):
\[ z + \frac{1}{z} = re^{i\theta} + \frac{1}{re^{i\theta}} = re^{i\theta} + \frac{1}{r} e^{-i\theta} \]
This simplifies to:
\[ |z + \frac{1}{z}| = |re^{i\theta} + \frac{1}{r} e^{-i\theta}| \]

### Step 3: Apply the Triangle Inequality
Using the triangle inequality, we have:
\[ |z + \frac{1}{z}| \leq |z| + |\frac{1}{z}| = r + \frac{1}{r} \]
Thus, we can write:
\[ |z + \frac{1}{z}| \leq r + \frac{1}{r} \]
Given that \( |z + \frac{1}{z}| = a \), we can conclude:
\[ a \leq r + \frac{1}{r} \]

### Step 4: Analyze the Function \( r + \frac{1}{r} \)
The function \( r + \frac{1}{r} \) achieves its minimum value when \( r = 1 \). Specifically, using calculus or the AM-GM inequality, we find:
\[ r + \frac{1}{r} \geq 2 \]
This means \( a \) must be at least 2 for there to be a solution.

### Step 5: Set Up the Quadratic Inequality
From the earlier step, we have:
\[ r + \frac{1}{r} \geq a \]
Multiplying through by \( r \) (assuming \( r > 0 \)):
\[ r^2 - ar + 1 \geq 0 \]
This is a quadratic inequality in \( r \).

### Step 6: Find the Roots of the Quadratic
The roots of the quadratic \( r^2 - ar + 1 = 0 \) can be found using the quadratic formula:
\[ r = \frac{a \pm \sqrt{a^2 - 4}}{2} \]
To ensure that the quadratic is non-negative, we need to consider the maximum value of \( r \).

### Step 7: Determine the Maximum Distance
The maximum distance from the origin occurs at the larger root:
\[ r_{\text{max}} = \frac{a + \sqrt{a^2 - 4}}{2} \]

### Conclusion
Thus, the maximum distance from the origin to the point \( z \) satisfying the equation \( |z + \frac{1}{z}| = a \) is:
\[ \boxed{\frac{a + \sqrt{a^2 - 4}}{2}} \]