Two circle , both of radii a , touch each other and each of them touches internally a circle of radius 2a , then the radiusof the circle which touches all the three circles is :

Options

A. `1/2 a`
B. `2/3 a`
C. `3/4 a`
D. a

Answer

To solve the problem of finding the radius of the circle that touches all three given circles, we can follow these steps:

### Step 1: Understand the Configuration
We have two smaller circles (C1 and C2) with radius \( a \) that touch each other. They also touch internally a larger circle (C3) with radius \( 2a \). We need to find the radius \( R \) of a smaller circle (C4) that touches all three circles.

### Step 2: Set Up the Geometry
1. **Draw the circles**:
- Circle C1 and C2 both have radius \( a \) and touch each other.
- Circle C3 has a radius of \( 2a \) and touches both C1 and C2 internally.

2. **Label the centers**:
- Let the center of C1 be \( A_1 \), the center of C2 be \( A_2 \), and the center of C3 be \( A \).
- Let the center of the smaller circle (C4) be \( A_3 \).

### Step 3: Use the Pythagorean Theorem
1. **Find distances**:
- The distance between \( A_1 \) and \( A_2 \) (the centers of circles C1 and C2) is \( 2a \) (since they touch each other).
- The distance from \( A \) (center of C3) to \( A_1 \) is \( 2a - a = a \) (since C3 touches C1 internally).
- The distance from \( A \) to \( A_2 \) is also \( a \).

2. **Set up the equation**:
- The distance \( A_1 A_3 \) is \( a + R \) (since C4 touches C1).
- The distance \( A_2 A_3 \) is \( a + R \) (since C4 touches C2).
- The distance \( A_1 A_2 \) is \( 2a \).

### Step 4: Apply Pythagorean Theorem
Using the triangle formed by \( A_1, A_2, A_3 \):
\[ (A_1 A_3)^2 + (A_2 A_3)^2 = (A_1 A_2)^2 \]
Substituting the distances:
\[ (a + R)^2 + (a + R)^2 = (2a)^2 \]
This simplifies to:
\[ 2(a + R)^2 = 4a^2 \]
Dividing both sides by 2:
\[ (a + R)^2 = 2a^2 \]

### Step 5: Solve for \( R \)
Taking the square root:
\[ a + R = a\sqrt{2} \]
Thus:
\[ R = a\sqrt{2} - a = a(\sqrt{2} - 1) \]

### Final Answer
The radius \( R \) of the circle that touches all three circles is:
\[ R = a(\sqrt{2} - 1) \]