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Focal distances of a point on an ellipse

This type of question, asked 4 times: 24 Jan 2025, 2 Apr 2025, 4 Apr 2025, 22 Jan 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    24 Jan 2025

    60
    Let the product of the focal distances of the point (3,12)\left( \sqrt{3},\frac{1}{2} \right) on the ellipse x2a2+y2 b2=1,(a>b)\frac{x^{2}}{a^{2}}+\frac{y^{2}}{{\text{ }b}^{2}}= 1,(a > b), be 74\frac{7}{4}. Then the absolute difference of the eccentricities of two such ellipses is

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  2. Question 2

    2 Apr 2025

    66
    If SS and S′S^{'} are the foci of the ellipse x218+y29=1\frac{x^{2}}{18}+\frac{y^{2}}{9}= 1 and P be a point on the ellipse, then min⁡(SP.S′P)+\min\left( SP.S^{'}P \right)+ max⁡(SP.S′P)\max\left( SP.S^{'}P \right) is equal to :

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  3. Question 3

    4 Apr 2025

    142
    The Centre of a circle C is at the Centre of the ellipse E:x2a2+y2b2=1,a>bE:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}= 1,a>b. Let CC pass through the foci F1F_{1} and F2F_{2} of EE such that the circle CC and the ellipse E intersect at four points. Let P be one of these four points. If the area of the triangle PF1 F2PF_{1}{\text{ }F}_{2} is 30 and the length of the major axis of E is 17 , then the distance between the foci of E is :

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  4. Question 4

    22 Jan 2026 S2

    66
    Let SS and S′S^{'} be the foci of the ellipse x225+y29=1\frac{x^{2}}{25}+\frac{y^{2}}{9}= 1 and P(α,β)P(\alpha,\beta) be a point on the ellipse in the first quadrant. If (SP)2+(S′P)2−SP∙S′P=37(SP)^{2}+\left( S^{'}P \right)^{2}- SP \bullet S^{'}P = 37, then α2+β2\alpha^{2}+\beta^{2} is equal to :

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