Let circle C₂ be the image of the circle C₁: x^2 + y^2 - 4x - 6y - 12 = 0 in the line x - 2y + 9 = 0 . Let A be the lowest point on C₂ . If point B lies on C₂ such that the length of the minor arc AB is (1/6)^ th of the perimeter of C₂ and the x -coordinate of B is positive, then the tangents to C₂ at points A and B intersect at a point P( , ) . The value of 3 + is

Step-by-step solution

The equation of circle C₁ is (x-2)^2 + (y-3)^2 = 25 . Its centre is (2, 3) and its radius is r = 5 . Let the centre of C₂ be (h, k) . Since C₂ is the reflection of C₁ in the line x - 2y + 9 = 0 , we have: h-2 1 = k-3 -2 = -2(2 - 2(3) + 9) 1^2 + (-2)^2 = -2(5) 5 = -2 h - 2 = -2 h = 0 k - 3 = 4 k = 7 The centre of C₂ is (0, 7) and its radius is 5 . The equation of C₂ is x^2 + (y-7)^2 = 25 . Point A is the lowest point on C₂ , so its coordinates are (0, 7 - 5) = (0, 2) . The parametric angle for A is - 2 . The length