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Locus of the midpoints of chords of a circle

This type of question, asked 2 times: 24 Jan 2023, 1 Feb 2024. All 2 questions below, oldest first.

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  1. Question 1

    24 Jan 2023

    156
    The locus of the mid points of the chords of the circle C1:(x−4)2+(y−5)2=4C_{1}:(x - 4)^{2}+ (y - 5)^{2}= 4 which subtend an angle θ1\theta_{1} at the centre of the circle C1C_{1}, is a circle of radius rir_{i}. If θ1=π3,θ3=2π3\theta_{1}=\frac{\pi}{3},\theta_{3}=\frac{2\pi}{3} and r12=r22+r32r_{1}^{2}=r_{2}^{2}+r_{3}^{2}, then θ2\theta_{2} is equal to

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  2. Question 2

    1 Feb 2024

    170
    Let the locus of the mid points of the chords of circle x2+(y−1)2=1x^{2}+ (y - 1)^{2}= 1 drawn from the origin intersect the line x+y=1x + y = 1 at PP and QQ. Then, the length of PQPQ is:

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