the locus of `z=i+2exp(i(theta+pi/4))` is

Options

A. A circle
B. An ellipse
C. A parabola
D. A hyperbola

Answer

To find the locus of the complex number \( z = i + 2e^{i(\theta + \frac{\pi}{4})} \), we will follow these steps:

### Step 1: Express \( z \) in terms of real and imaginary parts
We start with the given equation:
\[ z = i + 2e^{i(\theta + \frac{\pi}{4})} \]
We can express \( z \) as:
\[ z = x + iy \]
Substituting this into the equation gives:
\[ x + iy = i + 2e^{i(\theta + \frac{\pi}{4})} \]

### Step 2: Rewrite the exponential term
Using Euler's formula, we can rewrite the exponential:
\[ e^{i(\theta + \frac{\pi}{4})} = \cos\left(\theta + \frac{\pi}{4}\right) + i\sin\left(\theta + \frac{\pi}{4}\right) \]
Thus, we have:
\[ z = i + 2\left(\cos\left(\theta + \frac{\pi}{4}\right) + i\sin\left(\theta + \frac{\pi}{4}\right)\right) \]
This simplifies to:
\[ z = i + 2\cos\left(\theta + \frac{\pi}{4}\right) + 2i\sin\left(\theta + \frac{\pi}{4}\right) \]
Combining the real and imaginary parts:
\[ z = 2\cos\left(\theta + \frac{\pi}{4}\right) + i\left(1 + 2\sin\left(\theta + \frac{\pi}{4}\right)\right) \]

### Step 3: Identify real and imaginary parts
From the above expression, we can identify:
\[ x = 2\cos\left(\theta + \frac{\pi}{4}\right) \]
\[ y = 1 + 2\sin\left(\theta + \frac{\pi}{4}\right) \]

### Step 4: Use trigonometric identities
We know that:
\[ \cos\left(\theta + \frac{\pi}{4}\right) = \frac{x}{2} \]
\[ \sin\left(\theta + \frac{\pi}{4}\right) = \frac{y - 1}{2} \]
Using the identity \( \cos^2 A + \sin^2 A = 1 \):
\[ \left(\frac{x}{2}\right)^2 + \left(\frac{y - 1}{2}\right)^2 = 1 \]

### Step 5: Simplify the equation
Squaring both sides gives:
\[ \frac{x^2}{4} + \frac{(y - 1)^2}{4} = 1 \]
Multiplying through by 4:
\[ x^2 + (y - 1)^2 = 4 \]

### Conclusion
This is the equation of a circle with center at \( (0, 1) \) and radius \( 2 \).

### Final Answer
The locus of \( z \) is a circle given by:
\[ x^2 + (y - 1)^2 = 2^2 \]