Solving the Complex Number Equation for \(x^2 + y^2\)

We are given a relationship between a complex number \(x + iy\) and a square root involving other complex numbers \(a + ib\) and \(c + id\). The goal is to find the value of \(x^2 + y^2\). This expression, \(x^2 + y^2\), is directly related to the modulus of the complex number \(x + iy\).

Understanding the Modulus of a Complex Number

For any complex number \(z = x + iy\), where \(x\) and \(y\) are real numbers representing the real and imaginary parts respectively, the modulus of \(z\) is denoted by \(|z|\) and is defined as \(|z| = \sqrt{x^2 + y^2}\). Squaring the modulus gives us \(|z|^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2\). Therefore, finding \(x^2 + y^2\) is equivalent to finding the square of the modulus of the complex number \(x + iy\).

Applying Modulus to the Given Equation

The given equation is:

\(x + iy = \sqrt {\frac{{a + ib}}{{c + id}}}\)

Let's take the modulus on both sides of this complex number equation:

\(|x + iy| = \left|\sqrt {\frac{{a + ib}}{{c + id}}}\right|\)

Using Modulus Properties

We use the property that for a complex number \(w\), \(|\sqrt{w}| = \sqrt{|w|}\). Let \(w = \frac{a + ib}{c + id}\). Then the right side becomes \(\sqrt{\left|\frac{{a + ib}}{{c + id}}\right|}\).

So, we have:

\(|x + iy| = \sqrt{\left|\frac{{a + ib}}{{c + id}}\right|}\)

Next, we use the property that for two complex numbers \(z_1\) and \(z_2\), \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\). Applying this to the expression inside the square root:

\(|x + iy| = \sqrt{\frac{|a + ib|}{|c + id|}}}\)

Calculating Moduli of \(a + ib\) and \(c + id\)

The modulus of \(a + ib\) is \(|a + ib| = \sqrt{a^2 + b^2}\), and the modulus of \(c + id\) is \(|c + id| = \sqrt{c^2 + d^2}\), where \(a, b, c, d\) are real numbers.

Substituting these into the equation for \(|x + iy|\):

\(|x + iy| = \sqrt{\frac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}}}}\)

This can be rewritten by combining the square roots:

\(|x + iy| = \sqrt{\sqrt{\frac{a^2 + b^2}{c^2 + d^2}}}}\)

Which simplifies to:

\(|x + iy| = \left(\frac{a^2 + b^2}{c^2 + d^2}\right)^{1/4}\)

Let's re-evaluate the modulus of the square root. If \(z = \sqrt{w}\), then \(|z|^2 = |w|\). This is the key property. Let's use this directly.

Given \(x + iy = \sqrt{\frac{a + ib}{c + id}}\). Let \(z = x + iy\) and \(w = \frac{a + ib}{c + id}\). So \(z = \sqrt{w}\).

We want to find \(x^2 + y^2\), which is \(|x + iy|^2 = |z|^2\).

Since \(z = \sqrt{w}\), \(z^2 = w\). Taking the modulus on both sides of \(z^2 = w\):

\(|z^2| = |w|\)

Using the property \(|z^2| = |z|^2\):

\(|z|^2 = |w|\)

Substituting \(z = x + iy\) and \(w = \frac{a + ib}{c + id}\):

\(|x + iy|^2 = \left|\frac{a + ib}{c + id}\right|\)

Now, apply the quotient property of modulus: \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\)

\(|x + iy|^2 = \frac{|a + ib|}{|c + id|}}\)

Substitute the modulus values \(|a + ib| = \sqrt{a^2 + b^2}\) and \(|c + id| = \sqrt{c^2 + d^2}\):

\(|x + iy|^2 = \frac{\sqrt{a^2 + b^2}}{\sqrt{c^2 + d^2}}}\)

Combine the square roots in the numerator and denominator:

\(|x + iy|^2 = \sqrt{\frac{a^2 + b^2}{c^2 + d^2}}}\)

Since \(|x + iy|^2 = x^2 + y^2\), we get:

\(x^2 + y^2 = \sqrt{\frac{a^2 + b^2}{c^2 + d^2}}}\)

Summary of Steps for finding \(x^2 + y^2\):

1. Recognize that \(x^2 + y^2\) is the square of the modulus of the complex number \(x + iy\).
2. Take the modulus of both sides of the given equation \(x + iy = \sqrt{\frac{{a + ib}}{{c + id}}}\).
3. Use the property \(|\sqrt{w}| = \sqrt{|w|}\) or more effectively, \(|z^2| = |z|^2\), by squaring the original equation implicitly through the modulus. If \(z = \sqrt{w}\), then \(|z|^2 = |w|\).
4. Apply the property \(\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}\).
5. Calculate the modulus of the numerator \(|a + ib|\) and the denominator \(|c + id|\).
6. Substitute these values to find the final expression for \(x^2 + y^2\).

This step-by-step process, leveraging the properties of complex numbers and their modulus, leads directly to the solution.

The value of \(x^2 + y^2\) is \(\sqrt{\frac{a^2 + b^2}{c^2 + d^2}}}\).