Let the circle C₂ : x^2 + y^2 - 18x - 16y + k = 0 be the mirror image of the circle C₁ : x^2 + y^2 - 2x - 4y - 20 = 0 in a line L . If d is the perpendicular distance from the origin to the line L , then the value of k + d is ________.

Step-by-step solution

The center of circle C₁ is O₁(1, 2) and its radius is r₁ = 1^2 + 2^2 - (-20) = 25 = 5 . The center of circle C₂ is O₂(9, 8) and its radius is r₂ = 9^2 + 8^2 - k = 145 - k . Since C₂ is the mirror image of C₁ , their radii must be equal. Therefore, 145 - k = 5 145 - k = 25 k = 120 . The line of reflection L is the perpendicular bisector of the line segment joining the centers O₁(1, 2) and O₂(9, 8) . The midpoint of O₁O₂ is M ( 1+9 2 , 2+8 2 ) = (5, 5) . The slope of O₁O₂ is m = 8-2 9-1 = 6 8 = 3 4 . The slope of the