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Two chords from a point on a circle bisected by a given line

This type of question, asked 2 times: 31 Jan 2023, 4 Apr 2026 S1. All 2 questions below, oldest first.

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  1. Question 1

    31 Jan 2023

    168
    The set of all values of a2a^{2} for which the line x+y=0x + y = 0 bisects two distinct chords drawn from a point P(1+a2,1−a2)P\left( \frac{1 + a}{2},\frac{1 - a}{2} \right) on the circle 2x2+2y2−(1+a)x−(1−a)y=02x^{2}+ 2y^{2}- (1 + a)x - (1 - a)y = 0, is equal to:

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  2. Question 2

    4 Apr 2026 S1

    63
    Suppose the two chords, drawn from the point (1, 2) on the circle x2+y2+x−3y=0x^{2}+y^{2}+x- 3y= 0 are bisected by the y -axis. If the other ends of these chords are R and S , and the midpoint of the line segment RS is (α,β)(\alpha,\beta), then 6(α+β)6(\alpha+\beta) is equal to :

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