If `x+iy=sqrt((a+ib)/(c+id))`, then find the value of `x^(2)+y^(2)`.

Answer

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

### Step 1: Take the modulus of both sides
We start by taking the modulus of both sides of the equation:
\[ |x + iy| = \left| \sqrt{\frac{a + ib}{c + id}} \right| \]

### Step 2: Use the property of modulus
We know that the modulus of a complex number \( z = x + iy \) is given by \( |z| = \sqrt{x^2 + y^2} \). Therefore, we can write:
\[ \sqrt{x^2 + y^2} = \left| \sqrt{\frac{a + ib}{c + id}} \right| \]

### Step 3: Simplify the right side
Using the property of modulus for square roots, we have:
\[ \left| \sqrt{\frac{a + ib}{c + id}} \right| = \sqrt{\left| \frac{a + ib}{c + id} \right|} = \sqrt{\frac{|a + ib|}{|c + id|}} \]

### Step 4: Calculate the modulus of the complex numbers
Now, we calculate the modulus of \( a + ib \) and \( c + id \):
\[ |a + ib| = \sqrt{a^2 + b^2} \]
\[ |c + id| = \sqrt{c^2 + d^2} \]

### Step 5: Substitute back into the equation
Substituting these values back, we get:
\[ \sqrt{x^2 + y^2} = \sqrt{\frac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}}} \]

### Step 6: Square both sides
Now, squaring both sides to eliminate the square root gives:
\[ x^2 + y^2 = \frac{a^2 + b^2}{c^2 + d^2} \]

### Final Result
Thus, the value of \( x^2 + y^2 \) is:
\[ \boxed{\frac{a^2 + b^2}{c^2 + d^2}} \]

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