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Chord of a parabola subtending a right angle at the vertex

This type of question, asked 2 times: 22 Jan 2026 S1, 8 Apr 2026 S2. All 2 questions below, oldest first.

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  1. Question 1

    22 Jan 2026 S1

    67
    If the chord joining the points P1(x1,y1)P_{1}\left( x_{1},y_{1} \right) and P2(x2P_{2}\left( x_{2} \right., y2y_{2} ) on the parabola y2=12xy^{2}= 12x subtends a right angle at the vertex of the parabola, then x1x2−y1y2x_{1}x_{2}-y_{1}y_{2} is equal to

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  2. Question 2

    8 Apr 2026 S2

    61
    Let O be the vertex of the parabola y2=4xy^{2}= 4x and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C , then the length of its latus rectum is :

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