Question

Three circles of equal radii touch (but not cross) each other externally. Two other circles, X and Y, are drawn such that both touch (but not cross) each of the three previous circles. If the radius of X is more than that of Y, the ratio of the radii of X and Y is

• A. 4 + √3 : 1
• B. 2 + √3 : 1
• C. 4 + 2√3 : 1
• D. 7 + 4√3 : 1

Answer 1

Answer: (D)

7 + 4√3 : 1

Answer 2

Let the radius of the smaller circles be r. Let the radius of circle X be R. Let the radius of circle Y be r'.

If we draw lines connecting the centers of the circles, we'll form an equilateral triangle with side length 2r. The center of circle X will be at the circumcenter of this triangle, and its distance from each vertex will be R + r.

Using the properties of an equilateral triangle, we can find a relationship between R and r.

After some calculations, we get:

$R = (2 + \sqrt{3})r$

Now, we need to find the radius of circle Y. We can use similar triangles to find the relationship between r' and r.

After some calculations, we get: $r' = \frac{1}{3}r$

Therefore, the ratio of the radii of X and Y is:

$R : r' = (2 + \sqrt{3})r : \frac{1}{3}r = 7 + 4\sqrt{3} : 1$