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GATE 2022 IN – Question 50

Engineering Mathematics · Complex variables · 2 marks · Multiple select

For $z=(a+jb)/(a-jb)$ with $a>0,b>0$, which statements are true?

  1. The phase is $2\tan^{-1}(b/a)$
  2. The phase is $\tan^{-1}(2b/a)$
  3. The magnitude is 1
  4. The magnitude is $\sqrt{(a^2+b^2)/(a^2-b^2)}$

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Show answer and explanation

Correct answer: (A) The phase is $2\tan^{-1}(b/a)$; (C) The magnitude is 1

Explanation

Numerator and denominator are conjugates with equal magnitude, so |z|=1. Their phases are +atan(b/a) and −atan(b/a); quotient phase is 2atan(b/a).

Thus **A,C**.