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Hyperbola and ellipse with a theta-dependent coefficient and related eccentricities

This type of question, asked 2 times: 1 Feb 2024, 28 Jan 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    1 Feb 2024

    73
    For 0<θ<π/20 < \theta < \pi/2, if the eccentricity of the hyperbola x2−y2csc⁡2θ=5x^{2}-y^{2}\csc^{2}\theta = 5 is 7\sqrt{7} times eccentricity of the ellipse x2csc⁡2θ+y2=5x^{2}\csc^{2}\theta +y^{2}= 5, then the value of θ\theta is :

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  2. Question 2

    28 Jan 2026 S1

    72
    For some θ∈(0,π2)\theta \in \left( 0,\frac{\pi}{2} \right), let the eccentricity and the length of the latus rectum of the hyperbola x2−y2sec⁡2θ=8x^{2} - y^{2}\sec^{2}\theta = 8 be e1e_{1} and l1\mathcal{l}_{1}, respectively, and let the eccentricity and the length of the latus rectum of the ellipse x2sec⁡2θ+y2=6x^{2}\sec^{2}\theta + y^{2} = 6 be e2e_{2} and l2\mathcal{l}_{2}, respectively. If e12=e22(sec⁡2θ+1)e_{1}^{2} = e_{2}^{2}\left( \sec^{2}\theta + 1 \right), then (l1l2e1e2)tan⁡2θ\left( \frac{\mathcal{l}_{1}\mathcal{l}_{2}}{e_{1}e_{2}} \right)\tan^{2}\theta is equal to ____\_\_\_\_ .