If z is any complex number satisfying | z – 1 | = 1, then wh... | MATH - CONJUGATE, MODULUS AND ARGUMENT OF COMPLEX NUMBERS | SolveeIt | Solveeit

Today, August 19

Question

If z is any complex number satisfying | z – 1 | = 1, then which of the following is correct?

A

arg (z – 1) = 2 arg z

B

2arg (z) = 2/3 arg (z² – z)

C

arg (z – 1) = arg (z + 1)

D

arg (z) = 2 arg (z + 1)

Answer

arg (z – 1) = 2 arg z

Explanation

Solution

Sol. Let z – 1 = e^iq Ž z = 1 + cos q + i sin q

= 2 cos²(q/2) + 2i

sin(q/2) cos (q/2)z = 2 cos (q/2) (cos (q/2) + i sin (q/2))

arg (z) = q/2 = 1/2 arg (z – 1)