If `|w|=2`, then the set of points `x+iy=w-(1)/(w)` lie on

Options

A. Circle
B. Ellipse
C. Parabola
D. Hyperbola

Answer

To solve the problem, we need to analyze the expression \( x + iy = w - \frac{1}{w} \) given that \( |w| = 2 \).

### Step-by-Step Solution:

1. **Understanding the modulus condition**:
Since \( |w| = 2 \), we know that \( w \) can be expressed as \( w = 2e^{i\theta} \) for some angle \( \theta \). This means \( w \) lies on a circle of radius 2 in the complex plane.

2. **Expressing \( w \)**:
Let \( w = 2(\cos \theta + i \sin \theta) \). Then, we can write:
\[ w = 2\cos \theta + 2i\sin \theta \]

3. **Finding \( \frac{1}{w} \)**:
We need to find \( \frac{1}{w} \):
\[ \frac{1}{w} = \frac{1}{2(\cos \theta + i \sin \theta)} = \frac{1}{2} \cdot \frac{\cos \theta - i \sin \theta}{\cos^2 \theta + \sin^2 \theta} = \frac{1}{2}(\cos \theta - i \sin \theta) \]

4. **Calculating \( w - \frac{1}{w} \)**:
Now we can compute \( w - \frac{1}{w} \):
\[ w - \frac{1}{w} = \left( 2\cos \theta + 2i\sin \theta \right) - \left( \frac{1}{2}(\cos \theta - i \sin \theta) \right) \]
Simplifying this gives:
\[ = 2\cos \theta + 2i\sin \theta - \frac{1}{2}\cos \theta + \frac{1}{2}i\sin \theta \]
\[ = \left( 2\cos \theta - \frac{1}{2}\cos \theta \right) + \left( 2i\sin \theta + \frac{1}{2}i\sin \theta \right) \]
\[ = \frac{4}{2}\cos \theta - \frac{1}{2}\cos \theta + \left( \frac{4}{2}i\sin \theta + \frac{1}{2}i\sin \theta \right) \]
\[ = \frac{3}{2}\cos \theta + \frac{5}{2}i\sin \theta \]

5. **Identifying \( x \) and \( y \)**:
From the above expression, we can identify:
\[ x = \frac{3}{2}\cos \theta, \quad y = \frac{5}{2}\sin \theta \]

6. **Finding the relationship between \( x \) and \( y \)**:
To find the relationship between \( x \) and \( y \), we can use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ \left( \frac{2x}{3} \right)^2 + \left( \frac{2y}{5} \right)^2 = 1 \]
Simplifying this gives:
\[ \frac{4x^2}{9} + \frac{4y^2}{25} = 1 \]
Multiplying through by 225 (the least common multiple of 9 and 25):
\[ 100x^2 + 36y^2 = 225 \]

7. **Final equation**:
The equation \( 100x^2 + 36y^2 = 225 \) represents an ellipse.

### Conclusion:
The set of points \( x + iy = w - \frac{1}{w} \) lies on an ellipse.