The image contains a mathematical problem involving complex numbers and algebraic expressions. The text reads:

1. $\frac{a^2 + b^2}{c^2 + d^2} = \frac{p^2}{q^2}$
2. Prove that $\frac{x^2 + y^2}{(x - iy)} \left( \frac{a - ib}{c - id} \right) = \frac{p - iq}{c - id}$

The image also states a condition:

• If $x - iy = \sqrt{(c - id)}$

Let me walk you through the approach to solving this.

Step-by-Step Guide:

1. Step 1: Break down the components.

   • You are given that $x - iy = \sqrt{c - id}$. This suggests that you need to square this expression to eliminate the square root.
   • So, squaring both sides gives:
     $(x - iy)^2 = c - id$

2. Step 2: Substitute into the equation.

   • Now substitute this value of $(x - iy)$ into the second expression and simplify.

3. Step 3: Work with complex numbers.

   • You will need to work through both sides using properties of complex numbers, particularly using the forms $a + bi$ for complex numbers, where $i$ is the imaginary unit.

4. Step 4: Handle the fractions carefully.

   • Simplify both sides by handling the algebra carefully, especially with the complex number operations such as conjugates if necessary.

5. Step 5: Prove the equality.

   • After performing the necessary substitutions and algebraic manipulations, verify that both sides simplify to the same expression.

Would you like more specific steps or further clarification on any part of the process?

Further Questions:

1. What is the role of $x - iy = \sqrt{(c - id)}$ in the proof?
2. How does working with complex numbers $a - ib$ and $c - id$ affect the algebraic manipulations?
3. What strategies can you use to simplify expressions involving complex numbers?
4. How do you handle squaring the complex number $x - iy$?
5. How does the condition $\frac{a^2 + b^2}{c^2 + d^2} = \frac{p^2}{q^2}$ influence the rest of the proof?

Tip: When working with complex numbers, remember that multiplying by the conjugate can help simplify denominators in expressions involving division.