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Locus arg((z - a)/(z - b)) = theta is an arc of a circle

This type of question, asked 4 times: 2021 Paper 18, 2021 Paper 22, 26 Jun 2022, 29 Jun 2022. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 18

    71
    The equation arg(z−1z+1)=π4arg\left( \frac{z - 1}{z + 1} \right)=\frac{\pi}{4} represents a circle with:

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  2. Question 2

    2021 Paper 22

    82
    A point zz moves in the complex plane such that arg(z−2z+2)=π4arg\left( \frac{z - 2}{z + 2} \right)=\frac{\pi}{4}, then the minimum value of ∣z−92−2i∣2|z- 9\sqrt{2}- 2i|^{2} is equal to

  3. Question 3

    26 Jun 2022

    62
    Let A={z∈C:∣z+1z−1<1∣}A =\left\{ z \in C:\left| \frac{z + 1}{z - 1}< 1 \right| \right\} and B={z∈C:arg(z−1z+1)=2π3}B =\left\{ z \in C:arg\left( \frac{z - 1}{z + 1} \right)=\frac{2\pi}{3} \right\}. Then A∩BA \cap B is:

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  4. Question 4

    29 Jun 2022

    152
    Let arg(z)arg(z) represent the principal argument of the complex number zz. Then, ∣z∣=3|z| = 3 and arg(z−1)−arg(z - 1) - arg(z+1)=π4arg(z + 1) =\frac{\pi}{4} intersect:

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