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Which curve a given line touches

This type of question, asked 2 times: 2021 Paper 2, 2021 Paper 22. All 2 questions below, oldest first.

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  1. Question 1

    2021 Paper 2

    73
    For which of the following curves, the line x+3y=23x+\sqrt{3}y= 2\sqrt{3} is the tangent at the point (332,12)\left( \frac{3\sqrt{3}}{2},\frac{1}{2} \right) ?

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  2. Question 2

    2021 Paper 22

    80
    The line 12xcosθ+5ysinθ=6012xcos\theta + 5ysin\theta = 60 is tangent to which of the following curves?

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