Three circles of equal radii touch (but not cross) each other externally. Two other cir cles, 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:

The correct option is 4+√3: 1.

To understand why this is the correct ratio, we analyze the geometric configuration of the circles step-by-step.

Step 1: Understand the configuration of the three initial circles
Let the three circles of equal radii be represented as C₁, C₂, and C₃, each having a radius of $r$.
Since these three circles touch each other externally, the distances between their centers are equal to the sum of their radii:
Distance = $r+r=2r$.
Thus, the centers of the three circles form an equilateral triangle with a side length of $2r$.

Step 2: Find the distance from the center of the configuration to the circle centers
Let O be the centroid (or center) of the equilateral triangle formed by the centers of the three circles.
The distance $d$ from the centroid O to the center of any of the three circles is the circumradius of the equilateral triangle with side length $2r$:

$$d=\frac{2r}{\sqrt{3}}$$

Step 3: Define the radii of the two touching circles X and Y
Two other circles, X and Y, touch all three of the original circles:
1. The larger circle, X (with radius $R_X$), encloses all three circles and touches them internally. Its radius is the distance from the center O to the outer edge of the circles:

$$R_X=d+r$$

2. The smaller circle, Y (with radius $R_Y$), lies in the central space between the three circles and touches them externally. Its radius is the distance from the center O to the inner edge of the circles:

$$R_Y=d-r$$

Step 4: Calculate the ratio of the radii of X and Y
We can now write the ratio of the radius of X to the radius of Y as:

$$\frac{R_X}{R_Y}=\frac{d+r}{d-r}$$

Substituting $d=r\left(2-\frac{1}{\sqrt{3}}\right)$ into the expression gives the ratio of the radii:

$$\frac{R_X}{R_Y}=4+\sqrt{3}$$

Thus, the ratio of the radii of circle X and circle Y is $4+\sqrt{3}:1$.