As shown in the figure, you are given a unit circle centered at , and unit circle centered at . You also have circles and which are reflections of and about the -axis. Find the radius of the circumscribed circle that is tangent to all four circles and has them inside it, and submit
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As the system is symmetrical about the y -axis, we can expect that the center of the circle tangent to the four circles lies on the y -axis. Let the center be P ( 0 , y 0 ) . Since the P is equidistance from the circumferences of the four circles, it must also be equidistance from the four centers. As the system is symmetrical about the y -axis, we need only to consider one side and I am choosing C 1 and C 2 and we have:
( 2 5 ) 2 + y 0 2 4 2 5 ⟹ y 0 = ( 4 5 ) 2 + ( y 0 + 1 ) 2 = 1 6 2 5 + 2 y 0 + 1 = 3 2 5 9
Then the radius of the circle tangent internally to the four circles is R = A P + 1 = ( 2 5 ) 2 + ( 3 2 5 9 ) 2 + 1 ≈ 4 . 1 0 6 3 5 0 6 0 1 9 9 ⟹ ⌊ 1 0 0 0 R ⌋ = 4 1 0 6 .