The equation of the circle which is touched by y = x, has its centre on the positive direction of the x-axis and cuts off a chord of length 2 units along the line 3y−x=0, is

Since the required circle has its centre on x-axis, So, let the coordinates of the centre be (a, 0). The circle touches y = x. Therefore,Radius = Length of the . from (a, 0) on x -y = 0⇒  Radius =a2The circle cuts off a chord of length 2 units along x−3y=0∴ a22=12+a−3×012+(3)22⇒a22=1+a24⇒a=2Thus, centre of the circle is at (2, 0) and radius =a2=2Hence, its equation is x2+y2−4x+2=0