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Common chord of two circles

This type of question, asked 2 times: 30 Jan 2024, 5 Apr 2024. All 2 questions below, oldest first.

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  1. Question 1

    30 Jan 2024

    173
    Consider two circles C1:x2+y2=25C_{1}:x^{2}+y^{2}= 25 and C2:(x−C_{2}:(x - α)2+y2=16\alpha)^{2}+y^{2}= 16, where α∈(5,9)\alpha \in (5,9). Let the angle between the two radii (one to each circle) drawn from one of the intersection points of C1C_{1} and C2C_{2} be sin⁡−1(638)\sin^{- 1}\left( \frac{\sqrt{63}}{8} \right). If the length of common chord of C1C_{1} and C2C_{2} is β\beta, then the value of (αβ)2(\alpha\beta)^{2} equals

  2. Question 2

    5 Apr 2024

    162
    Let the circle C1:x2+y2−2(x+y)+1=0C_{1}:x^{2}+y^{2}- 2(x + y) + 1 = 0 and C2C_{2} be a circle having centre at (−1,0)( - 1,0) and radius 2. If the line of the common chord of C1C_{1} and C2C_{2} intersects the yy-axis at the point PP, then the square of the distance of PP from the centre of C1C_{1} is:

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