Let C be a circle passing through the points A(4,5) and B(6,1) . The radius of the circle C is given as 5 . If the centre of C lies on the circle x^2 + y^2 - 10x - 6y + k = 0 , then the value of k is equal to

Step-by-step solution

The centre of the circle C must lie on the perpendicular bisector of the line segment AB . The midpoint of AB is ( 4+6 2 , 5+1 2 ) = (5,3) . The slope of AB is 1-5 6-4 = -2 . The slope of the perpendicular bisector is 1 2 . The equation of the perpendicular bisector is y - 3 = 1 2 (x - 5) x - 2y + 1 = 0 . Let the centre of C be (2 - 1, ) . The distance from the centre to the point A(4,5) is equal to the radius, which is 5 . (2 - 1 - 4)^2 + ( - 5)^2 = 5^2 (2 - 5)^2 + ( - 5)^2 = 25 4 ^2 - 20 + 25 + ^2 - 10 + 25 = 25