There are two circles `C_(1)` and `C_(2)` touching each other and the coordinate axes, if `C_(1)` is smaller than `C_(2)` and its radius is 2 units, then radius of `C_(2)`, is

Options

A. `6+4sqrt(2)`
B. `2+2sqrt(2)`
C. `3+2sqrt(2)`
D. none of these

Explanation

To solve the problem of finding the radius of the second circle \( C_2 \), we will follow these steps:

### Step 1: Understand the Configuration of the Circles
We have two circles \( C_1 \) and \( C_2 \) that are touching each other and also touching the coordinate axes. The radius of the smaller circle \( C_1 \) is given as 2 units.

### Step 2: Set Up the Coordinates
Let the radius of the larger circle \( C_2 \) be \( r \). The center of circle \( C_1 \) will be at the point \( (2, 2) \) since it touches both axes. The center of circle \( C_2 \) will be at the point \( (r, r) \).

### Step 3: Calculate the Distance Between the Centers
The distance \( OP \) from the origin \( O(0, 0) \) to the center of circle \( C_1 \) is calculated as follows:
\[ OP = \sqrt{(2 - 0)^2 + (2 - 0)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]

### Step 4: Calculate the Distance from the Origin to the Center of Circle \( C_2 \)
The distance \( OQ \) from the origin \( O(0, 0) \) to the center of circle \( C_2 \) is:
\[ OQ = \sqrt{(r - 0)^2 + (r - 0)^2} = \sqrt{r^2 + r^2} = \sqrt{2r^2} = r\sqrt{2} \]

### Step 5: Relate the Distances
Since the circles are touching each other, the distance \( OQ \) can be expressed as the sum of the distance \( OP \) and the radii of both circles:
\[ OQ = OP + PR + RQ \]
Where:
- \( PR \) is the radius of circle \( C_1 \) which is 2 units.
- \( RQ \) is the radius of circle \( C_2 \) which is \( r \).

Thus, we have:
\[ r\sqrt{2} = 2\sqrt{2} + 2 + r \]

### Step 6: Rearrange the Equation
Rearranging gives:
\[ r\sqrt{2} - r = 2\sqrt{2} + 2 \]
Factoring out \( r \) from the left side:
\[ r(\sqrt{2} - 1) = 2\sqrt{2} + 2 \]

### Step 7: Solve for \( r \)
Now, we can solve for \( r \):
\[ r = \frac{2\sqrt{2} + 2}{\sqrt{2} - 1} \]

### Step 8: Rationalize the Denominator
To rationalize the denominator, multiply the numerator and denominator by \( \sqrt{2} + 1 \):
\[ r = \frac{(2\sqrt{2} + 2)(\sqrt{2} + 1)}{(\sqrt{2} - 1)(\sqrt{2} + 1)} = \frac{(2\sqrt{2} + 2)(\sqrt{2} + 1)}{2 - 1} = (2\sqrt{2} + 2)(\sqrt{2} + 1) \]

### Step 9: Expand the Numerator
Expanding the numerator:
\[ = 2\sqrt{2} \cdot \sqrt{2} + 2\sqrt{2} + 2\cdot\sqrt{2} + 2 = 4 + 4\sqrt{2} + 2 = 6 + 4\sqrt{2} \]

### Final Answer
Thus, the radius of the second circle \( C_2 \) is:
\[ r = 6 + 4\sqrt{2} \]