The distance between the centres of the circles which pass through A ( a , a ) and B ( 2 a , 2 a ) and touch the y-axis is

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The centre lies on the perpendicular bisector of AB , x + y = 3 a , so it is ( α , 3 a − α ) and the radius is ∣ α ∣ . Passing through A: ( α − a ) 2 + ( 2 a − α ) 2 = α 2 , i.e. α 2 − 6 a α + 5 a 2 = 0 . The roots α , β satisfy α + β = 6 a , α β = 5 a 2 , so ∣ α − β ∣ = 4 a . The centres are ( α , 3 a − α ) and ( β , 3 a − β ) , so C 1 ​ C 2 ​ = 2 ​ ∣ α − β ∣ = 4 2 ​ a .

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