ω is a complex cube root of unity and Z is a complex number satisfying |Z−1|≤2. The possible values of r such that |Z−1|≤2 and |ωZ−1−ω2|=r have no common solution are
We have, |z−1|≤2 ‌ And ‌‌|ωz−1−ω2|=r ‌|ω||z−ω2−ω|=r ‌|z+1|=r ‌[∵|ω|=1,1+ω+ω2=0] ‌|z−1|≤2⇒|z+1−2|≤2 ‌|z+1|−2≤2⇒|z+1|≤4 For no solution ∴‌‌|z+1|>4⇒r>4