Disc cut by a line: greatest or least distance of the region from a point
This type of question, asked 5 times: 2021 Paper 17, 2021 Paper 26, 14 Jun 2022, 29 Jun 2022, 1 Feb 2024. All 5 questions below, oldest first.
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Question 1
2021 Paper 17
65
Let C be the set of all complex numbers. Let S1={z∈C:∣z−2∣≤1} and S2={z∈C:z(1+i)+z(1−i)≥4}. Then, the maximum value of z−252 for z∈S1∩S2 is equal to:
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Question 2
2021 Paper 26
89
Let z be those complex number which satisfy ∣z+5∣≤4 and z(1+i)+zˉ(1−i)≥−10,i=−1. If the maximum value of ∣z+1∣2 is α+β2, then the value of (α+β) is
Question 3
14 Jun 2022
171
Let S={z∈C:∣z−3∣≤1 and z(4+3i)+zˉ(4−3i)≤24}. If α+iβ is the point in S which is closest to 4i, then 25(α+β) is equal to
Question 4
29 Jun 2022
81
Let S={z∈C:∣z−2∣≤1,z(1+i)+zˉ(1− i) ≤2}. Let ∣z−4i∣ attains minimum and maximum values, respectively, at z1∈S and z2∈S. If 5(∣z1∣2+∣z2∣2)=α+β5, where α and β are integers, then the value of α+β is equal to
Question 5
1 Feb 2024
87
Let P={z∈C:∣z+2−3i∣≤1} and Q={z∈C:z(1+i)+z(1−i)≤−8}. Let in P∩Q,∣z−3+2i∣ be maximum and minimum at z1 and z2 respectively. If ∣z1∣2+2∣z2∣2=α+β2, where α,β are integers, then α+β equals
