A circle of radius `5u n i t s` touches the coordinate axes in the first quadrant. If the circle makes one complete roll on x-axis along he positive direction of x-axis , find its equation in new position.

Answer

To solve the problem step by step, let's break down the process of finding the new equation of the circle after it rolls along the x-axis.

### Step 1: Identify the center of the original circle
The circle has a radius of 5 units and touches both the x-axis and y-axis in the first quadrant. Therefore, the center of the circle is at:
\[ (5, 5) \]

### Step 2: Determine the distance covered in one complete roll
When the circle rolls one complete revolution on the x-axis, the distance covered is equal to the circumference of the circle. The formula for the circumference \(C\) of a circle is:
\[ C = 2\pi r \]
Substituting the radius \(r = 5\):
\[ C = 2\pi \times 5 = 10\pi \text{ units} \]

### Step 3: Calculate the new center after rolling
After rolling one complete revolution, the new center of the circle will be shifted to the right by the distance of the circumference. Therefore, the new center coordinates will be:
\[ (5 + 10\pi, 5) \]

### Step 4: Write the equation of the new circle
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 \(h = 5 + 10\pi\), \(k = 5\), and \(r = 5\):
\[ (x - (5 + 10\pi))^2 + (y - 5)^2 = 5^2 \]
This simplifies to:
\[ (x - (5 + 10\pi))^2 + (y - 5)^2 = 25 \]

### Step 5: Expand the equation
Now, we will expand the equation:
\[ (x - (5 + 10\pi))^2 = (x - 5 - 10\pi)^2 = (x - 5)^2 - 2(x - 5)(10\pi) + (10\pi)^2 \]
Thus, the equation becomes:
\[ (x - 5)^2 - 20\pi(x - 5) + 100\pi^2 + (y - 5)^2 = 25 \]

### Step 6: Combine and simplify
Combining everything, we have:
\[ (x - 5)^2 + (y - 5)^2 - 20\pi(x - 5) + 100\pi^2 = 0 \]
This is the new equation of the circle after rolling along the x-axis.

### Final Answer
The equation of the circle in its new position is:
\[ (x - (5 + 10\pi))^2 + (y - 5)^2 = 25 \]

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