Question

Let z and ω be complex numbers satisfying zz̅+ω\(\overline{\omega}\)=100. If |z|,|ω|∈I, then find the minimum possible value of the expression |z+ω|.

(a) 1

(b) 2

(c) 3

(d) 4

I had been asked this question in an internship interview.

Asked question is from Complex Conjugates in section Complex Numbers of Engineering Mathematics

Answer 1

Right choice is (b) 2

Easiest explanation: zz̅+ω\(\overline{\omega}\)=100⇒|z|^2+|ω|^2=100, also, since |z|, |ω|∈I^+, therefore, (|z|,|ω|)=(8,6) or (6,8). Hence, |z+ω|≥||z|-|ω||=|8-6|=2.