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

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

Accepted Answer

### The figure will be as shown below:

![image.png](https://res.cloudinary.com/aptidude/image/upload/v1742489349/aptidude_questions/nw0t5smfqldctn70ocpi.png)

Let’s take the radius of each of the three inscribed medium-sized circles to be **R**, and that of the circle **Y** (in between these three circles) to be **r**.

If we join the centers of the three medium-sized circles, we will get an equilateral triangle of length **2R**. The distance between these circle's centers and the smallest circle's **Y** center will be **(R + r)**, as shown below.

---

### Step 1: Geometry of the Triangle

Now, we draw a radius from the center of the medium-sized circle to the point where it touches the other medium-sized circle. We connect this point of contact with the center of the smallest circle **Y**.

This gives us a **30°–60°–90°** right-angled triangle, as shown below.

---

### Step 2: Calculations

From the geometry, we find:

$$
R + r = R \cdot \frac{2}{\sqrt{3}}
$$

$$
r = R \cdot \frac{(2\sqrt{3} - 3)}{3}.
$$

The biggest circle **X** will have a radius as:

$$
2R + r = R \cdot \left[ 2 + \frac{(2\sqrt{3} - 3)}{3} \right] = R \cdot \frac{(2\sqrt{3} + 3)}{3}.
$$

---

### Step 3: Ratio of Radii

We are asked to find the ratio of radii of circles **X** and **Y**, which is:

$$
\frac{(2\sqrt{3} + 3)}{(2\sqrt{3} - 3)}.
$$

Simplify:

$$
= (2\sqrt{3} + 3) : (2\sqrt{3} - 3)
$$

Multiply by conjugates:

$$
= (21 + 12\sqrt{3}) : (7 + 4\sqrt{3}).
$$

---

### Final Answer:

The ratio is:

$$7 + 4\sqrt{3} : 1.$$

Answer

Answer

Answer