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Mirror image of a circle in a line equals a given circle: find the unknown constant and the radius

The same question, asked 2 times: 29 Jul 2022*, 8 Apr 2023. Every wording below, oldest first.

Try each one first, then open its answer.

  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

    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