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Least or greatest distance from a point or circle to a disc, ring or circle

This type of question, asked 4 times: 1 Feb 2024, 29 Jan 2025, 23 Jan 2026 S1, 2 Apr 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    1 Feb 2024

    162
    If zz is a complex number such that ∣z∣≥1|z| \geq 1, then the minimum value of ∣z+12(3+4i)∣\left| z +\frac{1}{2}(3 + 4i) \right| is:

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  2. Question 2

    29 Jan 2025

    69
    Let ∣z1−8−2i∣≤1\left| z_{1}- 8 - 2i \right|\leq 1 and ∣z2−2+6i∣≤2\left| z_{2}- 2 + 6i \right|\leq 2, z1,z2∈Cz_{1},z_{2}\in C. Then the minimum value of ∣z1−z2∣\left| z_{1}-z_{2} \right| is :

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  3. Question 3

    23 Jan 2026 S1

    67
    Let S={z:3≤∣2z−3(1+i)∣≤7}\mathbf{S = \{ z:3 \leq |2z - 3(1 + i)| \leq 7\}} be a set of complex numbers. Then Minz∈s∣(z+12(5+3i))∣\underset{z \in s}{Min}\left| \left( z +\frac{1}{2}(5 + 3i) \right) \right| is equal to :

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  4. Question 4

    2 Apr 2026 S2

    52
    Let the circle C1:∣z∣=rC_{1}:|z| = r and C2:∣z−3−4i∣=5,z∈CC_{2}:|z - 3 - 4i| = 5,z \in\mathbb{C}, be such that C2C_{2} lies within C1C_{1}. If z1z_{1} moves on C1,z2C_{1},z_{2} moves on C2C_{2} and min⁡∣z1−z2∣=2\min\left| z_{1}-z_{2} \right|= 2, then max ∣z1−z2∣\left| z_{1}-z_{2} \right| is equal to :-

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