If `x+iy=3/(2+costheta +i sin theta)`, then show that `x^2+y^2=4x-3`

Answer

To solve the problem, we need to show that if \( x + iy = \frac{3}{2 + \cos \theta + i \sin \theta} \), then \( x^2 + y^2 = 4x - 3 \).

### Step-by-Step Solution

**Step 1: Start with the given equation.**
\[ x + iy = \frac{3}{2 + \cos \theta + i \sin \theta} \]

**Step 2: Rationalize the denominator.**
To simplify the right-hand side, multiply the numerator and the denominator by the conjugate of the denominator:
\[ x + iy = \frac{3(2 + \cos \theta - i \sin \theta)}{(2 + \cos \theta)^2 + \sin^2 \theta} \]
Here, the denominator simplifies as follows:
\[ (2 + \cos \theta)^2 + \sin^2 \theta = 4 + 4\cos \theta + \cos^2 \theta + \sin^2 \theta = 4 + 4\cos \theta + 1 = 5 + 4\cos \theta \]

**Step 3: Substitute back into the equation.**
Now we can rewrite \( x + iy \):
\[ x + iy = \frac{3(2 + \cos \theta - i \sin \theta)}{5 + 4\cos \theta} \]
This gives us:
\[ x = \frac{3(2 + \cos \theta)}{5 + 4\cos \theta}, \quad y = \frac{-3\sin \theta}{5 + 4\cos \theta} \]

**Step 4: Calculate \( x^2 + y^2 \).**
Now we need to find \( x^2 + y^2 \):
\[ x^2 + y^2 = \left(\frac{3(2 + \cos \theta)}{5 + 4\cos \theta}\right)^2 + \left(\frac{-3\sin \theta}{5 + 4\cos \theta}\right)^2 \]
\[ = \frac{9(2 + \cos \theta)^2 + 9\sin^2 \theta}{(5 + 4\cos \theta)^2} \]
\[ = \frac{9((2 + \cos \theta)^2 + \sin^2 \theta)}{(5 + 4\cos \theta)^2} \]

**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:
\[ x^2 + y^2 = \frac{9(5 + 4\cos \theta)}{(5 + 4\cos \theta)^2} = \frac{9}{5 + 4\cos \theta} \]

**Step 6: Calculate \( 4x - 3 \).**
Now calculate \( 4x - 3 \):
\[ 4x - 3 = 4\left(\frac{3(2 + \cos \theta)}{5 + 4\cos \theta}\right) - 3 \]
\[ = \frac{12(2 + \cos \theta)}{5 + 4\cos \theta} - 3 = \frac{12(2 + \cos \theta) - 3(5 + 4\cos \theta)}{5 + 4\cos \theta} \]
\[ = \frac{24 + 12\cos \theta - 15 - 12\cos \theta}{5 + 4\cos \theta} = \frac{9}{5 + 4\cos \theta} \]

**Step 7: Conclusion.**
We have shown that:
\[ x^2 + y^2 = \frac{9}{5 + 4\cos \theta} = 4x - 3 \]
Thus, we conclude that:
\[ x^2 + y^2 = 4x - 3 \]