Let the mirror image of the circle C₁ : x^2 + y^2 - 4x - 6y + 12 = 0 in the line L : x - y + c = 0 be the circle C₂ . If C₂ touches the line 4x + 3y - 10 = 0 , then the maximum possible value of c is ______

Step-by-step solution

The given circle is C₁ : x^2 + y^2 - 4x - 6y + 12 = 0 . The center of C₁ is (2, 3) and its radius is r = 2^2 + 3^2 - 12 = 4 + 9 - 12 = 1 . Since C₂ is the mirror image of C₁ , the radius of C₂ is also 1 . Let the center of C₂ be (h, k) . It is the mirror image of (2, 3) in the line x - y + c = 0 . Using the reflection formula: h - 2 1 = k - 3 -1 = -2(2 - 3 + c) 1^2 + (-1)^2 h - 2 1 = k - 3 -1 = -(c - 1) = 1 - c h = 3 - c and k = 2 + c So, the center of C₂ is (3 - c, 2 + c) . Since C₂ touches the line 4x + 3y - 10 =