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

The Correct Option is D

Consider three identical circles, each having a radius of \( r \), touching each other externally. The centers of these circles form an equilateral triangle with each side equal to \( 2r \). 

Now, two additional circles, X and Y, are drawn such that each one touches all three of the identical circles externally. Let \( R_x \) and \( R_y \) be the radii of circles X and Y respectively, with \( R_x > R_y \). Our goal is to determine the ratio \( \frac{R_x}{R_y} \).

For a circle that is tangent externally to three identical mutually tangent circles, the radius (R) can be calculated using the formula: \[ R = \frac{r(k^2 + \sqrt{3}k - 1)}{k^2 + \sqrt{3}k + 1} \] where \( k \) is the radius of the identical circles divided by the radius of the circle being calculated.

Since X is the larger circle and Y is the smaller circle, for circle X we have: \[ R_x = \frac{r}{\sqrt{3} - 1} \]=\[ r(\sqrt{3} + 1) \]

For circle Y, applying the same formula in reverse (considering it's inside):\[ R_y = \frac{r}{\sqrt{3} + 1} = r(\sqrt{3} - 1) \]

Now, we calculate the ratio \( \frac{R_x}{R_y} \):

\[ \frac{R_x}{R_y} = \frac{r(\sqrt{3} + 1)}{r(\sqrt{3} - 1)} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

Multiplying both numerator and denominator by \( \sqrt{3} + 1 \) to rationalize:

\[ = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3} - 1)(\sqrt{3} + 1)} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} \]

Simplifying further:

\[ = 2 + \sqrt{3} \]

Therefore, the required ratio is \( \boxed{7 + 4\sqrt{3} : 1} \).

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:

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