A circle of radius `2` units touches the co ordinate axes in the first quadrant. If the circle makes a complete rotation on the x-axis along the positive direction of the x-axis, then the equation of the circle in the new position is

Options

A. `x^(2)+y^(2)-4(x+y)-8pi x + (4pi + 2)^(2)=0`
B. `x^(2)+y^(2)-4x-4y+(4 pi + 2)^(2)=0`
C. `x^(2)+y^(2)-8 pi x - 4y+(4 pi + 2)^(2)=0`
D. none of these

Explanation

To solve the problem step by step, we need to find the equation of the circle after it makes a complete rotation around the x-axis. Here’s how we can do it:

### Step 1: Identify the initial position of the circle
The circle has a radius of 2 units and touches both the x-axis and y-axis in the first quadrant. Therefore, the center of the circle is at the point (2, 2).

### Step 2: Determine the circumference of the circle
The circumference \( C \) of a circle is given by the formula:
\[ C = 2\pi r \]
Substituting the radius \( r = 2 \):
\[ C = 2\pi \times 2 = 4\pi \]

### Step 3: Find the new center after rotation
When the circle makes a complete rotation along the positive direction of the x-axis, the center of the circle will move forward by the distance equal to its circumference.

The initial center is (2, 2). After moving forward by \( 4\pi \), the new center \( (x', y') \) will be:
\[ x' = 2 + 4\pi \]
\[ y' = 2 \]
Thus, the new center is \( (4\pi + 2, 2) \).

### Step 4: Write the equation of the circle in its new position
The general equation of a circle with center \( (h, k) \) and radius \( r \) is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 4\pi + 2 \), \( k = 2 \), and \( r = 2 \):
\[ (x - (4\pi + 2))^2 + (y - 2)^2 = 2^2 \]
This simplifies to:
\[ (x - (4\pi + 2))^2 + (y - 2)^2 = 4 \]

### Step 5: Expand the equation
Now, we will expand the equation:
1. Expanding \( (x - (4\pi + 2))^2 \):
\[ (x - (4\pi + 2))^2 = x^2 - 2(4\pi + 2)x + (4\pi + 2)^2 \]
2. Expanding \( (y - 2)^2 \):
\[ (y - 2)^2 = y^2 - 4y + 4 \]
3. Combining both expansions:
\[ x^2 - 2(4\pi + 2)x + (4\pi + 2)^2 + y^2 - 4y + 4 = 4 \]
4. Simplifying:
\[ x^2 + y^2 - 2(4\pi + 2)x - 4y + (4\pi + 2)^2 + 4 - 4 = 0 \]
\[ x^2 + y^2 - 2(4\pi + 2)x - 4y + (4\pi + 2)^2 = 0 \]

### Final Equation
The final equation of the circle in its new position is:
\[ x^2 + y^2 - (8\pi + 4)x - 4y + (4\pi + 2)^2 = 0 \]