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Points for which a line segment subtends an angle of pi/4 at the origin

The same question, asked 2 times: 29 Jun 2022*, 8 Apr 2024. Every wording below, oldest first.

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  1. Question 1

    29 Jun 2022

    73
    The distance between the two points AA and A′A^{'} which lie on y=2y = 2 such that both the line segments ABAB and A′BA^{'}B (where BB is the point (2,3)(2,3)) subtend angle π4\frac{\pi}{4} at the origin, is equal to:

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  2. Question 2

    8 Apr 2024

    162
    If the line segment joining the points (5,2)(5,2) and (2, a) subtends an angle π4\frac{\pi}{4} at the origin, then the absolute value of the product of all possible values of aa is:

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