Let a circle C₂ be the mirror image of the circle C₁ : x^2 + y^2 - 6x - 2y + 6 = 0 in the line 2x - y - 10 = 0 . If C₂ intersects the circle C₃ : x^2 + y^2 + 2x + 2ky - 34 = 0 orthogonally, then the value of k is ________.

Step-by-step solution

The center of circle C₁ is O₁(3, 1) and its radius is r₁ = 3^2 + 1^2 - 6 = 4 = 2 . Let the center of circle C₂ be O₂(h, k') . Since O₂ is the reflection of O₁ in the line 2x - y - 10 = 0 , we have: h - 3 2 = k' - 1 -1 = -2 2(3) - 1(1) - 10 2^2 + (-1)^2 h - 3 2 = k' - 1 -1 = -2 ( -5 5 ) = 2 Solving for h and k' : h - 3 = 4 h = 7 k' - 1 = -2 k' = -1 So, the center of C₂ is O₂(7, -1) . Since reflection preserves the radius, the radius of C₂ is r₂ = 2 . The equation of C₂ is: (x - 7)^2 + (y + 1)^2 = 2^2 x^2 - 14x + 49