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Tangent to an ellipse meeting the axes: minimum intercept sum, area or length

This type of question, asked 3 times: 2021 Paper 10, 2021 Paper 20, 24 Jan 2023. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    2021 Paper 10

    75
    Let a tangent be drawn to the ellipse x227+y2=1\frac{x^{2}}{27}+y^{2}= 1 at (33cos⁡θ,sin⁡θ)(3\sqrt{3}\cos\theta,\sin\theta) where θ∈(0,π2)\theta \in\left( 0,\frac{\pi}{2} \right). Then the value of θ\theta such that the sum of intercepts on axes made by this tangent is minimum is equal to:

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  2. Question 2

    2021 Paper 20

    88
    If the minimum area of the triangle formed by a tangent to the ellipse x2 b2+y24a2=1\frac{x^{2}}{{\text{ }b}^{2}}+\frac{y^{2}}{4a^{2}}= 1 and the co-ordinate axis is kabkab, then kk is equal to

  3. Question 3

    24 Jan 2023

    82
    Let a tangent to the curve 9x2+16y2=1449x^{2}+ 16y^{2}= 144 intersect the coordinate axes at the points AA and BB. Then, the minimum length of the line segment ABAB is