CAT — Base System

Given:
Three equal circles of radius r touch each other externally.
Two more circles, X and Y, are drawn such that each touches all the three circles.
Radius of X is greater than that of Y.
Find the ratio of the radii of X and Y.

Step 1: Locate the centres
The centres of the three equal circles form an equilateral triangle of side
2r.
The centre of both circles X and Y is the common centre (circumcentre) of this equilateral triangle.
The distance from this centre to each vertex is the circumradius of the equilateral triangle.
Circumradius
= (2r)/√3
= 2r/√3

Step 2: Find the radius of the larger circle X
Since X encloses the three circles, the distance between its centre and the centre of each small circle is
R − r,
where R is the radius of X.
Hence,
R − r = 2r/√3
R = r + 2r/√3
= r(1 + 2/√3)

Step 3: Find the radius of the smaller circle Y
Since Y lies inside the gap, the distance between its centre and the centre of each small circle is
r + y,
where y is the radius of Y.
Hence,
r + y = 2r/√3
y = 2r/√3 − r
= r(2/√3 − 1)

Step 4: Find the ratio
R : y
= (1 + 2/√3) : (2/√3 − 1)
Multiply both terms by √3,
= (√3 + 2) : (2 − √3)
Rationalizing,
= (√3 + 2)² : (4 − 3)
= (3 + 4 + 4√3) : 1
= (7 + 4√3) : 1

Answer:
A. (7 + 4√3) : 1