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Other end of a focal chord of a parabola from one end

This type of question, asked 3 times: 22 Jan 2025, 2 Apr 2025, 21 Jan 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    22 Jan 2025

    128
    Let P(4,43)P(4,4\sqrt{3}) be a point on the parabola y2=4axy^{2}= 4ax and PQPQ be a focal chord of the parabola. If MM and N are the foot of perpendiculars drawn from P and QQ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:

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  2. Question 2

    2 Apr 2025

    142
    Let the point PP of the focal chord PQPQ of the parabola y2=16xy^{2}= 16x be (1,−4)(1, - 4). If the focus of the parabola divides the chord PQ in the ratio m:nm:n, gcd(m,n)=1gcd(m,n) = 1, then m2+n2m^{2}+n^{2} is equal to :

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  3. Question 3

    21 Jan 2026 S2

    56
    Let one end of a focal chord of the parabola y2=16xy^{2}= 16x be (16, 16). If P(α,β)P(\alpha,\beta) divides this focal chord internally in the ratio 5:25:2, then the minimum value of α+β\alpha + \beta is equal to :

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