Question

For the complex number \( Z = \frac{a + jb}{a - jb} \), where \( a>0 \) and \( b>0 \). Which of the following statement(s) is/are true?

• A. The phase is \( 2 \tan^{-1} \frac{b}{a} \)
• B. The phase is \( \tan^{-1} \frac{2b}{a} \)
• C. The magnitude is 1
• D. The magnitude is \( \sqrt{\frac{a^2 + b^2}{a^2 - b^2}} \)

Hint:

For a complex number \( Z = \frac{a + jb}{a - jb} \), the phase is \( 2 \tan^{-1} \frac{b}{a} \), and the magnitude is 1.

Answer 1

The Correct Option is A, C

Step 1: The phase of a complex number.
The phase of the complex number \( Z = \frac{a + jb}{a - jb} \) can be found using the formula for the argument (angle) of a complex number. For this specific case, the phase \( \theta \) is: \[ \theta = \arg\left(\frac{a + jb}{a - jb}\right) = 2 \tan^{-1}\left(\frac{b}{a}\right) \] Thus, the phase is \( 2 \tan^{-1} \frac{b}{a} \), which makes option (A) correct. Step 2: The magnitude of a complex number.
The magnitude of the complex number \( Z = \frac{a + jb}{a - jb} \) is given by the ratio of the magnitudes of the numerator and denominator: \[ |Z| = \left|\frac{a + jb}{a - jb}\right| = \frac{|a + jb|}{|a - jb|} \] Since both the numerator and denominator have the same magnitude \( \sqrt{a^2 + b^2} \), the magnitude of \( Z \) is: \[ |Z| = 1 \] Therefore, the magnitude is 1, which makes option (C) correct. Step 3: Conclusion.
Thus, the correct answers are (A) and (C).