Question - Complex Number Division and Its Parts
Solution:
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 $$\frac{2x}{x^2 + 4}j$$.
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