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Number of points common to a circle and an ellipse given in complex form

The same question, asked 2 times: 27 Jun 2022*, 8 Apr 2026 S2. Every wording below, oldest first.

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  1. Question 1

    27 Jun 2022

    151
    ∣z−(4+3i)∣=2|z - (4 + 3i)| = 2 and ∣z∣+∣z−4∣=6,z∈C|z| + |z - 4| = 6,z \in C is :

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  2. Question 2

    8 Apr 2026 S2

    52
    The number of values of z∈Cz\in\mathbb{C}, satisfying the equations ∣z−(4+8i)∣=10|z- (4 + 8i)| =\sqrt{10} and ∣z−(3+5i)∣+∣z−(5+11i)∣=45|z- (3 + 5i)| + |z- (5 + 11i)| = 4\sqrt{5}, is :

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