The circles having radii 1, 2, 3 touch cach other externally then the radius of the circle which cuts three circles orthogonally is

Options

A. 1
B. 4
C. 6
D. 2

Answer

To solve the problem of finding the radius of a circle that cuts three circles with radii 1, 2, and 3 orthogonally, we can follow these steps:

### Step 1: Identify the given circles and their radii
We have three circles with the following radii:
- Circle A: \( r_1 = 1 \)
- Circle B: \( r_2 = 2 \)
- Circle C: \( r_3 = 3 \)

### Step 2: Calculate the distances between the centers of the circles
Since the circles touch each other externally, the distances between their centers can be calculated as follows:
- Distance \( AB = r_1 + r_2 = 1 + 2 = 3 \)
- Distance \( AC = r_1 + r_3 = 1 + 3 = 4 \)
- Distance \( BC = r_2 + r_3 = 2 + 3 = 5 \)

### Step 3: Calculate the semi-perimeter \( S \)
The semi-perimeter \( S \) of the triangle formed by the centers of the circles can be calculated using the formula:
\[ S = \frac{AB + AC + BC}{2} = \frac{3 + 4 + 5}{2} = \frac{12}{2} = 6 \]

### Step 4: Calculate the area \( \Delta \) of the triangle
Using Heron's formula, the area \( \Delta \) can be calculated as:
\[ \Delta = \sqrt{S(S - AB)(S - AC)(S - BC)} \]
Substituting the values:
\[ \Delta = \sqrt{6(6 - 3)(6 - 4)(6 - 5)} = \sqrt{6 \times 3 \times 2 \times 1} = \sqrt{36} = 6 \]

### Step 5: Use the formula to find the radius \( R \) of the orthogonal circle
The radius \( R \) of the circle that cuts the three circles orthogonally is given by the formula:
\[ R = \frac{\Delta}{S} \]
Substituting the values we found:
\[ R = \frac{6}{6} = 1 \]

### Conclusion
The radius of the circle that cuts the three circles orthogonally is \( R = 1 \).

### Final Answer
The radius of the circle which cuts the three circles orthogonally is **1**.