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Qno3:

• Separate $\frac{x+iy}{x-iy}$ into real and imaginary parts.

Let’s work on separating the real and imaginary components of the given complex expression:

Expression:

$\frac{x+iy}{x-iy}$

Step 1: Multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of $x - iy$ is $x + iy$. Multiplying both the numerator and denominator by this conjugate:

$\frac{x+iy}{x-iy} \times \frac{x+iy}{x+iy} = \frac{(x+iy)(x+iy)}{(x-iy)(x+iy)}$

Step 2: Simplify the denominator.

Using the difference of squares formula:

$(x-iy)(x+iy) = x^2 - (iy)^2 = x^2 - (-y^2) = x^2 + y^2$

So the denominator becomes $x^2 + y^2$.

Step 3: Simplify the numerator.

$(x+iy)(x+iy) = x^2 + 2ixy + i^2y^2 = x^2 + 2ixy - y^2$

Thus, the expression becomes: $\frac{x^2 - y^2 + 2ixy}{x^2 + y^2}$

Step 4: Separate into real and imaginary parts.

Now, split the fraction into real and imaginary components:

$\frac{x^2 - y^2}{x^2 + y^2} + i \frac{2xy}{x^2 + y^2}$

Thus, the real part is: $\frac{x^2 - y^2}{x^2 + y^2}$ And the imaginary part is: $\frac{2xy}{x^2 + y^2}$

This is the expression separated into real and imaginary parts.

Would you like further explanation or details on any step?

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Here are 5 related questions to expand on this concept:

1. How do you find the conjugate of a complex number?
2. What are the properties of the difference of squares used in simplifying complex expressions?
3. How would you express a complex number in polar form?
4. Can you provide an example of simplifying a different complex fraction?
5. What is the geometric interpretation of the division of two complex numbers?

Tip: Always check if you can multiply by a conjugate when simplifying fractions involving complex numbers. It makes the denominator real, which simplifies the expression.