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Arc of a circle subtending a right angle at the centre: write OC in terms of OA and OB

The same question, asked 2 times: 10 Apr 2023*, 23 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    10 Apr 2023

    80
    An arc PQPQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If OP→=u→,OR→=v→\overrightarrow{OP}=\overrightarrow{u},\overrightarrow{OR}=\overrightarrow{v} and OQ→=αu→+βv→\overrightarrow{OQ}= \alpha\overrightarrow{u}+ \beta\overrightarrow{v}, then α,β2\alpha,\beta^{2} are the roots of the equation

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  2. Question 2

    23 Jan 2025

    57
    Let the arc AC of a circle subtend a right angle at the centre O . If the point B on the arcACarcAC, divides the arcACarcAC such that  length of arcAB length of arcBC=15\frac{\text{~length of~}arcAB}{\text{~length of~}arcBC}=\frac{1}{5}, and OC→=αOA→+βOB→\overrightarrow{OC}= \alpha\overrightarrow{OA}+ \beta\overrightarrow{OB}, then α+2(3−1)β\alpha +\sqrt{2}(\sqrt{3}- 1)\beta is equal to

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