The locus of the middle points of the chords of the circle C 1 : x - 4 2 + y - 5 2 = 4 which subtend an angle θ i at the centre of the circle C i , is a circle of radius r i . If θ 1 = π 3 , θ 3 = 2 π 3 and r 1 2 = r 2 2 + r 3 2 , then θ 2 is equal to

Step-by-step solution

Given, C 1 :   x - 4 2 + y - 5 2 = 4 which subtend an angle θ i at the centre of the circle C i , is a circle of radius r i , Now plotting the diagram we get, Now given If θ 1 = π 3 , θ 3 = 2 π 3 and r 1 2 = r 2 2 + r 3 2 , Now taking triangle C P B we get, cos θ 2 = P C 2 , so P C = 2 cos θ 2 Now using distance formula we get, h - 4 2 + k - 5 2 = 4 cos 2 θ 2 So, locus will be, x - 4 2 + y - 5 2 = 4 cos 2 θ 2 Now radius r 1 = 2 cos π 6 = 3 , r 2 = 2 cos θ