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Right-angled triangle with tan of the two half-angles as roots of a quadratic: relation among the coefficients

The same question, asked 5 times: 11 May 2023 S2, 9 May 2024 S2, 19 Apr 2025 S2, 25 Apr 2025 S2, 26 Apr 2025 S2. Every wording below, oldest first.

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  1. Question 1

    11 May 2023 S2

    126
    In a △PQR\triangle PQR, m∠R=π2m\angle R = \frac{\pi}{2}. If tan⁡P2\tan\frac{P}{2} and tan⁡Q2\tan\frac{Q}{2} are the roots of the equation ax2+bx+c=0 (a≠0)ax^2+bx+c=0\ (a\neq0), then

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  2. Question 2

    9 May 2024 S2

    123
    In △ABC\triangle ABC, with usual notations, ∠C=π2\angle C = \frac{\pi}{2}, if tan⁡A2\tan\frac{A}{2} and tan⁡B2\tan\frac{B}{2} are the roots of the equation a1x2+b1x+c1=0  (a1≠0)a_1x^2 + b_1x + c_1 = 0\;(a_1 \neq 0), then

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  3. Question 3

    19 Apr 2025 S2

    118
    If triangle ABC is a right angled at A and tan⁡B2\tan\frac{B}{2}, tan⁡C2\tan\frac{C}{2} are roots of the equation ax2+bx+c=0ax^{2}+bx+c= 0, a≠0a\neq 0, then

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  4. Question 4

    25 Apr 2025 S2

    133
    With usual notations in △ABC\bigtriangleup ABC, if ∠B=π2\angle B =\frac{\pi}{2}, and tan⁡A2,tan⁡C2\tan\frac{A}{2},\tan\frac{C}{2} are roots of equation px2+qx+r=0px^{2}+ qx + r = 0, p≠0p\neq 0, then

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  5. Question 5

    26 Apr 2025 S2

    113
    In a triangle PQR with usual notations, ∠R=π2\angle R =\frac{\pi}{2}. If tan⁡P2\tan\frac{P}{2} and tan⁡Q2\tan\frac{Q}{2} are the roots of the equation ax2+bx+c=0(a≠0)ax^{2}+bx+c= 0(a\neq 0), then

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