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x sin t = y cos t with a second relation: eliminate t

This type of question, asked 3 times: CDS 2 2016, CDS 1 2017, CDS 2 2023. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 2 2016

    76
    If 2ycos⁡θ=xsin⁡θ2y\cos\theta = x\sin\theta and 2xsec⁡θ−ycosec⁡θ=32x\sec\theta - y\operatorname{cosec}\theta = 3, then what is x2+4y2x^2 + 4y^2 equal to ?

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  2. Question 2

    CDS 1 2017

    61
    If xsin⁡θ=ycos⁡θ=2ztan⁡θ1−tan⁡2θx \sin \theta = y \cos \theta = \frac{2z \tan \theta}{1 - \tan^2 \theta}, then what is 4z2(x2+y2)4z^2 (x^2 + y^2) equal to ?

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  3. Question 3

    CDS 2 2023

    30
    If xsin⁡3θ+ycos⁡3θ=sin⁡θcos⁡θx\sin^3\theta + y\cos^3\theta = \sin\theta\cos\theta and xsin⁡θ−ycos⁡θ=0x\sin\theta - y\cos\theta = 0, for every θ∈(0,π2)\theta \in \left(0, \dfrac{\pi}{2}\right), then what is x2+y2x^2 + y^2 equal to ?

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