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Chord of a parabola or ellipse with a given midpoint

This type of question, asked 8 times: 27 Jan 2024, 29 Jan 2024, 4 Apr 2024, 23 Jan 2025, 24 Jan 2025, 28 Jan 2025, 29 Jan 2025, 21 Jan 2026 S1. All 8 questions below, oldest first.

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  1. Question 1

    27 Jan 2024

    75
    The length of the chord of the ellipse x225+y216=1\frac{x^{2}}{25}+\frac{y^{2}}{16}= 1, whose mid point is (1,25)\left( 1,\frac{2}{5} \right), is equal to :

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  2. Question 2

    29 Jan 2024

    173
    Let P(α,β)P(\alpha,\beta) be a point on the parabola y2=4xy^{2}= 4x. If PP also lies on the chord of the parabola x2=8yx^{2}= 8y whose mid point is (1,54)\left( 1,\frac{5}{4} \right). Then (α−28)(β−8)(\alpha- 28)(\beta- 8) is equal to

  3. Question 3

    4 Apr 2024

    166
    Let PQPQ be a chord of the parabola y2=12xy^{2}=12x and the midpoint of PQPQ be at (4,1)(4,1). Then which of the following points lies on the line passing through the points PP and QQ?

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  4. Question 4

    23 Jan 2025

    140
    The length of the chord of the ellipse x24+y22=1\frac{x^{2}}{4}+\frac{y^{2}}{2}= 1 whose mid-point is (1,12)\left( 1,\frac{1}{2} \right), is:

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  5. Question 5

    24 Jan 2025

    126
    The equation of the chord, of the ellipse x225+y216=1\frac{x^{2}}{25}+\frac{y^{2}}{16}= 1, whose mid-point is (3,1)(3,1) is :

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  6. Question 6

    28 Jan 2025

    130
    If the midpoint of a chord of the ellipse x29+y24=1\frac{x^{2}}{9}+\frac{y^{2}}{4}= 1 is (2,4/3)(\sqrt{2},4/3), and the length of the chord is 2α3\frac{2\sqrt{\alpha}}{3}, then α\alpha is :

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  7. Question 7

    29 Jan 2025

    134
    If αx+βy=109\alpha x + \beta y = 109 is the equation of the chord of the ellipse x29+y24=1\frac{x^{2}}{9}+\frac{y^{2}}{4}= 1, whose mid point is (52,12)\left( \frac{5}{2},\frac{1}{2} \right), then α+β\alpha + \beta is equal to

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  8. Question 8

    21 Jan 2026 S1

    70
    Let O be the vertex of the parabola x2=4yx^{2}= 4y and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C , which is bisected at the point (1,2)(1,2), is:

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