If z and ω are two complex numbers such that | zω | = 1 and arg (z) - arg (ω ) = 3π/2, then arg ( (1 - 2z̅ ω)/(1 + 3z̅ ω) ) is :
(Here arg(z) denotes the principal argument of complex number z)

Solution

Maths sir · 4 steps

First pin down z̅ω as one exact number. Then the fraction is just ordinary arithmetic.

1. Modulus of the product. |z̅ω| = |z̅| |ω| = |z| |ω| = |zω| = 1. So z̅ω lies on the unit circle.
2. Argument of the product. Conjugating flips the sign of the argument, so arg(z̅ω) = -arg z + arg ω = -3π/2. That is the same direction as π/2, so z̅ω = i.

   (1-2i)/(1+3i) = (1-2i)(1-3i)/(1+3i)(1-3i) = (1-5i+6i²)/10 = (-5-5i)/10 = -1/2-1/2i

3. Find the quadrant. The real part and the imaginary part are both negative and equal, so the point is in the third quadrant at 45° from the negative real axis. Its principal argument is -π + π/4 = -3π/4.
4. So. arg((1-2z̅ω)/(1+3z̅ω)) = -3π/4, which is option (b).

The idea to keep. When a modulus and an argument difference are given, turn them into one exact complex number first. Remember that arg z̅ = -arg z, and always check the quadrant before you write the principal argument.

Answer from the NTA final answer key. Solution by PrepWiser.