If `x+iy= ( 3)/( 2 + cos theta +i sin theta )`, then `x^(2) + y^(2)` is equal to

Options

A. a) 3x-4
B. b) 4x-3
C. c) 4x+3
D. d) None of these

Answer

To solve the problem \( x + iy = \frac{3}{2 + \cos \theta + i \sin \theta} \) and find \( x^2 + y^2 \), we will follow these steps:

### Step 1: Rationalize the denominator
We start with the expression:
\[ x + iy = \frac{3}{2 + \cos \theta + i \sin \theta} \]
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ x + iy = \frac{3(2 + \cos \theta - i \sin \theta)}{(2 + \cos \theta + i \sin \theta)(2 + \cos \theta - i \sin \theta)} \]

### Step 2: Simplify the denominator
Now we simplify the denominator:
\[ (2 + \cos \theta)^2 + \sin^2 \theta \]
Calculating this:
\[ (2 + \cos \theta)^2 = 4 + 4\cos \theta + \cos^2 \theta \]
Thus, the denominator becomes:
\[ 4 + 4\cos \theta + \cos^2 \theta + \sin^2 \theta = 4 + 4\cos \theta + 1 = 5 + 4\cos \theta \]

### Step 3: Write the expression for \( x + iy \)
Now substituting back, we have:
\[ x + iy = \frac{3(2 + \cos \theta - i \sin \theta)}{5 + 4\cos \theta} \]
This can be separated into real and imaginary parts:
\[ x = \frac{3(2 + \cos \theta)}{5 + 4\cos \theta}, \quad y = \frac{-3\sin \theta}{5 + 4\cos \theta} \]

### Step 4: Calculate \( x^2 + y^2 \)
To find \( x^2 + y^2 \), we compute:
\[ x^2 = \left(\frac{3(2 + \cos \theta)}{5 + 4\cos \theta}\right)^2 \]
\[ y^2 = \left(\frac{-3\sin \theta}{5 + 4\cos \theta}\right)^2 \]
Thus,
\[ x^2 + y^2 = \frac{9(2 + \cos \theta)^2}{(5 + 4\cos \theta)^2} + \frac{9\sin^2 \theta}{(5 + 4\cos \theta)^2} \]
Combining these:
\[ x^2 + y^2 = \frac{9((2 + \cos \theta)^2 + \sin^2 \theta)}{(5 + 4\cos \theta)^2} \]

### Step 5: Simplify the numerator
Now simplify the numerator:
\[ (2 + \cos \theta)^2 + \sin^2 \theta = 4 + 4\cos \theta + \cos^2 \theta + \sin^2 \theta = 4 + 4\cos \theta + 1 = 5 + 4\cos \theta \]
Thus,
\[ x^2 + y^2 = \frac{9(5 + 4\cos \theta)}{(5 + 4\cos \theta)^2} \]

### Step 6: Final simplification
This simplifies to:
\[ x^2 + y^2 = \frac{9}{5 + 4\cos \theta} \]

### Conclusion
Thus, the final result is:
\[ x^2 + y^2 = \frac{9}{5 + 4\cos \theta} \]