Let-z-and-w-be-two-complex-numbers-such-that-z-1-w-1-and-z-iw-z-iw-2-then-find-the-value-of-z-

Question Number 230 by ssahoo last updated on 25/Jan/15

Let z and w be two complex numbers such that

∣ z ∣⩽ 1, ∣ w ∣⩽ 1 and ∣ z + i w ∣=∣ z - i \overset{―}{w} ∣= 2,

then find the value of z .

Commented by 123456 last updated on 16/Dec/14

∣ z + i w ∣⩽∣ z ∣ + ∣ w ∣⩽ 2

∣ z - i \bar{w} ∣⩽∣ z ∣ + ∣ \bar{w} ∣=∣ z ∣ + ∣ w ∣⩽ 2

∣ z + i w ∣=∣ z - i \bar{w} ∣= 2

∣ z ∣⩽ 1, ∣ w ∣⩽ 1

∣ z ∣=∣ w ∣= 1

z = 1, w = - i

z + i w = 1 - i^{2} = 1 + 1 = 2

z - i \bar{w} = 1 - i^{2} = 1 + 1 = 2

Answered by prakash jain last updated on 16/Dec/14

With the arugments in comments

z = 1 and w = - i

or z = - 1 and w = i

so the possible values of z are 1 and - 1