If `a + ib = sqrt((u + iv)/(x + iy))` then the value of `a^2 + b^2` is:

Options

A. `sqrt((u^2 + v^2)/(x^2 + y^2))`
B. `(u^2 - v^2)/(x^2 - y^2)`
C. can't be determined
D. None of these

Answer

To solve the problem \( a + ib = \sqrt{\frac{u + iv}{x + iy}} \) and find the value of \( a^2 + b^2 \), we can follow these steps:

### Step 1: Express the equation
We start with the equation:
\[ a + ib = \sqrt{\frac{u + iv}{x + iy}} \]

### Step 2: Multiply by the conjugate
To simplify the right-hand side, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{u + iv}{x + iy} \cdot \frac{x - iy}{x - iy} = \frac{(u + iv)(x - iy)}{(x + iy)(x - iy)} \]

### Step 3: Simplify the denominator
The denominator simplifies as follows:
\[ (x + iy)(x - iy) = x^2 + y^2 \]

### Step 4: Expand the numerator
Now, we expand the numerator:
\[ (u + iv)(x - iy) = ux - uiy + ivx + v(y) = ux + vy + i(vx - uy) \]

### Step 5: Combine results
Now, we can write:
\[ \frac{u + iv}{x + iy} = \frac{ux + vy + i(vx - uy)}{x^2 + y^2} \]

### Step 6: Take the square root
Taking the square root gives us:
\[ a + ib = \sqrt{\frac{ux + vy}{x^2 + y^2}} + i\sqrt{\frac{vx - uy}{x^2 + y^2}} \]

### Step 7: Identify \( a \) and \( b \)
From the above expression, we can identify:
\[ a = \sqrt{\frac{ux + vy}{x^2 + y^2}}, \quad b = \sqrt{\frac{vx - uy}{x^2 + y^2}} \]

### Step 8: Calculate \( a^2 + b^2 \)
Now, we calculate \( a^2 + b^2 \):
\[ a^2 + b^2 = \left(\sqrt{\frac{ux + vy}{x^2 + y^2}}\right)^2 + \left(\sqrt{\frac{vx - uy}{x^2 + y^2}}\right)^2 \]
\[ = \frac{ux + vy}{x^2 + y^2} + \frac{vx - uy}{x^2 + y^2} \]
\[ = \frac{(ux + vy) + (vx - uy)}{x^2 + y^2} \]

### Step 9: Combine the terms
Combining the terms in the numerator:
\[ = \frac{ux + vy + vx - uy}{x^2 + y^2} \]

### Step 10: Final result
Thus, we have:
\[ a^2 + b^2 = \frac{u^2 + v^2}{x^2 + y^2} \]

### Conclusion
The value of \( a^2 + b^2 \) is:
\[ \frac{u^2 + v^2}{x^2 + y^2} \]