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Mirror image of a circle in a line

This type of question, asked 4 times: 29 Jul 2022, 25 Jan 2023, 8 Apr 2023, 24 Jan 2025. All 4 questions below, oldest first.

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  1. Question 1

    29 Jul 2022

    90
    Let the mirror image of a circle c1:x2+y2−2x−c_{1}:x^{2}+y^{2}- 2x - 6y+α=06y + \alpha = 0 in line y=x+1y = x + 1 be c2:5x2+5y2+10gxc_{2}:5x^{2}+ 5y^{2}+ 10gx +10 fy +38=0+ 38 = 0. If rr is the radius of circle c2c_{2}, then α+6r2\alpha + 6r^{2} is equal to

  2. Question 2

    25 Jan 2023

    61
    Let the points of intersection of the line ax+by=0 (a≠b)ax+by=0\ (a\neq b) and the circle x2+y2−2x=0x^{2}+y^{2}-2x=0 be A(α,0)A(\alpha,0) and B(1,β)B(1,\beta). The image of the circle with ABAB as a diameter in the line x+y+2=0x+y+2=0 is:

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  3. Question 3

    8 Apr 2023

    87
    Consider a circle C1:x2+y2−4x−2y=α−5C_{1}:x^{2}+y^{2}- 4x - 2y = \alpha - 5. Let its mirror image in the line y=2x+1y = 2x + 1 be another circle C2:5x2+5y2−10fx−10gy+36=0C_{2}:5x^{2}+ 5y^{2}- 10fx - 10gy + 36 = 0. Let  r\ r be the radius of C2C_{2}. Then α+r\alpha + r is equal to

  4. Question 4

    24 Jan 2025

    65
    Let circle CC be the image of x2+y2−2x+4y−4=0x^{2}+y^{2}- 2x + 4y - 4 = 0 in the line 2x−3y+5=02x - 3y + 5 = 0 and AA be the point on CC such that OA is parallel to xx-axis and A lies on the right hand side of the centre OO of CC. If B(α,β)B(\alpha,\beta), with β<4\beta < 4, lies on CC such that the length of the arc AB is (1/6)th (1/6)^{\text{th~}} of the perimeter of C , then β−3α\beta -\sqrt{3}\alpha is equal to

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