Complex Number Division and Its Parts
The problem asks for the real and imaginary parts of the complex number \(\frac{z_1}{z_2}\), where \(z_1 = x + 4j\) and \(z_2 = x + 2j\).
To find \(\frac{z_1}{z_2}\), you can divide \(z_1\) by \(z_2\). Here's how you do it:
\[
\frac{z_1}{z_2} = \frac{x + 4j}{x + 2j}
\]
To divide two complex numbers, you can multiply the numerator and the denominator by the conjugate of the denominator:
\[
\frac{x + 4j}{x + 2j} \cdot \frac{x - 2j}{x - 2j} = \frac{(x + 4j)(x - 2j)}{x^2 - (2j)^2}
\]
Computing the products in the numerator and simplifying the denominator (using \(j^2 = -1\)):
\[
\frac{x^2 - 2jx + 4xj - 8j^2}{x^2 - 4j^2} = \frac{x^2 + 2xj - 8(-1)}{x^2 - 4(-1)}
\]
Simplify further:
\[
\frac{x^2 + 2xj + 8}{x^2 + 4}
\]
Now you have the complex number in fractional form. Let's separate it into real and imaginary parts:
The real part is:
\[
\frac{x^2 + 8}{x^2 + 4}
\]
The imaginary part is:
\[
\frac{2xj}{x^2 + 4}
\]
Hence, the real part of \(\frac{z_1}{z_2}\) is \(\frac{x^2 + 8}{x^2 + 4}\) and the imaginary part is \(\frac{2x}{x^2 + 4}j\).
