If z lies on unit circle with center at the origin, then `(1+z)/(1+barz)` is equal to

Options

A. z
B. `barz`
C. `z+barz`
D. none of these

Explanation

To solve the problem, we need to find the value of \(\frac{1+z}{1+\bar{z}}\) given that \(z\) lies on the unit circle. Here’s a step-by-step solution:

### Step 1: Understanding the Unit Circle
Since \(z\) lies on the unit circle, we know that the modulus of \(z\) is equal to 1. This means:
\[ |z| = 1 \]

### Step 2: Using the Property of Modulus
From the property of complex numbers, we have:
\[ |z|^2 = z \cdot \bar{z} \]
Given that \(|z| = 1\), we can square both sides:
\[ |z|^2 = 1 \implies z \cdot \bar{z} = 1 \]

### Step 3: Rewrite the Expression
We need to simplify the expression \(\frac{1+z}{1+\bar{z}}\). We can rewrite this as:
\[ \frac{1+z}{1+\bar{z}} = \frac{1+z}{1+\frac{1}{z}} \quad \text{(since } \bar{z} = \frac{1}{z} \text{ for } |z| = 1\text{)} \]

### Step 4: Simplifying the Denominator
Now, we can simplify the denominator:
\[ 1 + \bar{z} = 1 + \frac{1}{z} = \frac{z + 1}{z} \]

### Step 5: Substitute Back into the Expression
Substituting this back into our expression gives:
\[ \frac{1+z}{1+\bar{z}} = \frac{1+z}{\frac{z+1}{z}} = (1+z) \cdot \frac{z}{z+1} \]

### Step 6: Cancel Common Terms
Notice that \(1 + z\) and \(z + 1\) are the same, so we can cancel them:
\[ (1+z) \cdot \frac{z}{z+1} = z \]

### Conclusion
Thus, we find that:
\[ \frac{1+z}{1+\bar{z}} = z \]

### Final Answer
The value of \(\frac{1+z}{1+\bar{z}}\) is \(z\).

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