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Normal at any point of a parametric curve: fixed point or distance from the origin

This type of question, asked 2 times: 14 May 2024 S1, 22 Apr 2025 S2. All 2 questions below, oldest first.

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  1. Question 1

    14 May 2024 S1

    126
    Let the curve be represented by x=2(cos⁡t+tsin⁡t)x=2(\cos t+t\sin t), y=2(sin⁡t−tcos⁡t)y=2(\sin t-t\cos t). Then normal at any point 'tt' of the curve is at a distance of units from the origin.

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  2. Question 2

    22 Apr 2025 S2

    134
    The normal to the curve x=9(1+cos⁡θ)x= 9(1 +\cos\theta), y=9sin⁡θy= 9\sin\theta at θ\theta always passes through the fixed point

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