If z lies on the circle I z l = 1, then 2/z lies on

Options

A. a circle
B. an ellipse
C. a straight line
D. a parabola

Explanation

To solve the problem step by step, we will analyze the given condition and derive the necessary conclusions.

### Step 1: Understand the given condition
We know that the complex number \( z \) lies on the circle defined by \( |z| = 1 \). This means that the modulus of \( z \) is equal to 1.

### Step 2: Express \( z \) in terms of trigonometric functions
Since \( |z| = 1 \), we can express \( z \) using Euler's formula:
\[ z = \cos \theta + i \sin \theta \]
for some angle \( \theta \).

### Step 3: Find \( \frac{2}{z} \)
To find \( \frac{2}{z} \), we can substitute our expression for \( z \):
\[ \frac{2}{z} = \frac{2}{\cos \theta + i \sin \theta} \]
To simplify this, we multiply the numerator and the denominator by the conjugate of the denominator:
\[ \frac{2}{z} = \frac{2(\cos \theta - i \sin \theta)}{(\cos \theta + i \sin \theta)(\cos \theta - i \sin \theta)} \]
The denominator simplifies to:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]
Thus, we have:
\[ \frac{2}{z} = 2(\cos \theta - i \sin \theta) = 2 \cos \theta - 2 i \sin \theta \]

### Step 4: Let \( \frac{2}{z} = x + iy \)
Now, we can set:
\[ x = 2 \cos \theta \quad \text{and} \quad y = -2 \sin \theta \]

### Step 5: Square both equations
Now we square both equations:
\[ x^2 = (2 \cos \theta)^2 = 4 \cos^2 \theta \]
\[ y^2 = (-2 \sin \theta)^2 = 4 \sin^2 \theta \]

### Step 6: Add the squared equations
Adding these two equations gives:
\[ x^2 + y^2 = 4 \cos^2 \theta + 4 \sin^2 \theta \]
Factoring out the 4:
\[ x^2 + y^2 = 4(\cos^2 \theta + \sin^2 \theta) \]
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ x^2 + y^2 = 4 \cdot 1 = 4 \]

### Conclusion
This shows that \( \frac{2}{z} \) lies on the circle defined by:
\[ x^2 + y^2 = 4 \]
This is a circle with a radius of 2 centered at the origin (0, 0).

### Final Answer
Thus, \( \frac{2}{z} \) lies on a circle with radius 2.

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