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Points common to a circle and a line (or two circles) in complex form

This type of question, asked 6 times: 26 Jul 2022, 29 Jan 2023, 1 Feb 2024, 4 Apr 2025, 24 Jan 2026 S1, 28 Jan 2026 S1. All 6 questions below, oldest first.

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  1. Question 1

    26 Jul 2022

    152
    If z=x+iyz = x + iy satisfies ∣z∣−2=0|z| - 2 = 0 and ∣z−i∣−∣z+5i∣=0|z - i| - |z + 5i| = 0, then

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  2. Question 2

    29 Jan 2023

    178
    Let α=8−14i,A={z∈C:αz−αˉzˉz2−(zˉ)2−112i=1}\alpha = 8 - 14i,A =\left\{ z\in C:\frac{\alpha z -\bar{\alpha}\bar{z}}{z^{2}- (\bar{z})^{2}- 112i}= 1 \right\} and B={z∈C:∣z+3i∣=4}\mathbf{B = \{ z \in C:|z + 3i| = 4\}}. Then ∑z∈A∩B(Rez−Imz)\sum_{z \in A \cap B} (Rez - Imz) is equal to

  3. Question 3

    1 Feb 2024

    66
    Let S={z∈C:∣z−1∣=1\mathbf{S = \{ z \in C:|z - 1| = 1} and (2−1)(z+zˉ)−i(z−zˉ)=22}(\sqrt{2}- 1)(z +\bar{z}) - i(z -\bar{z}) = 2\sqrt{2}\}. Let z1, z2z_{1},\ z_{2} ∈S\in S be such that ∣z1∣=max⁡z∈S∣z∣\left| z_{1} \right|=\max_{z \in S} |z| and ∣z2∣=min⁡z∈S∣z∣\left| z_{2} \right|=\min_{z \in S} |z|. Then ∣2z1−z2∣2\left| \sqrt{2}z_{1}-z_{2} \right|^{2} equals:

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  4. Question 4

    4 Apr 2025

    73
    Let A={z∈C:∣z−2−i∣=3}A = \{ z \in C:|z - 2 - i| = 3\}, B={z∈C:Re(z−iz)=2}B = \{ z \in C:Re(z - iz) = 2\} and S=A∩BS = A \cap B. Then ∑z∈S∣z∣2\sum_{z \in S}^{} |z|^{2} is equal to .

  5. Question 5

    24 Jan 2026 S1

    59
    Let S={z∈C:∣z−6iz−2i∣=1S =\left\{ z\in C:\left| \frac{z - 6i}{z - 2i} \right|= 1 \right. and  ∣z−8+2iz+2i∣=35}\left. \ \left| \frac{z - 8 + 2i}{z + 2i} \right|=\frac{3}{5} \right\}. Then ∑z∈s∣z∣2\sum_{z \in s} |z|^{2} is equal to

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  6. Question 6

    28 Jan 2026 S1

    55
    Let zz be a complex number such that ∣z−6∣=5|z- 6| = 5 and ∣z+2−6i∣=5|z+ 2 - 6i| = 5. Then the value of z3+3z2−15z+141z^{3}+ 3z^{2}- 15z+ 141 is equal to

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