Finding the Equation of a Rolling Circle on the X-axis

Let's analyze the problem involving a circle rolling on the x-axis.

Initially, the circle has a radius of 5 units and touches the co-ordinate axes in the first quadrant. When a circle touches both the x and y axes in the first quadrant, its center is located at a point where the distance to the x-axis is equal to the radius and the distance to the y-axis is also equal to the radius. Since the radius is 5 units, the initial center of the circle is at (5, 5).

The equation of the circle in its initial position is given by the standard form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

Initial equation: $(x - 5)^2 + (y - 5)^2 = 5^2$.

Now, the circle rolls one complete roll on the x-axis along the positive direction. When a circle rolls along a straight line for one complete revolution, the distance covered by its center along that line is equal to the circumference of the circle.

The radius of the circle is $r = 5$ units.

The circumference of the circle is given by the formula $C = 2\pi r$.

Circumference = $2 \times \pi \times 5 = 10\pi$ units.

As the circle rolls one complete turn on the positive x-axis, its center moves a distance equal to its circumference in the positive x direction.

The initial coordinates of the center were $(5, 5)$.

• The x-coordinate of the center changes because the circle is rolling along the x-axis. The distance moved is $10\pi$. Since it rolls in the positive direction, the new x-coordinate will be the initial x-coordinate plus the distance rolled.
• New x-coordinate = Initial x-coordinate + Distance rolled
• New x-coordinate = $5 + 10\pi$.

• The y-coordinate of the center remains constant when a circle rolls on a horizontal line (like the x-axis) while maintaining contact. The distance of the center from the x-axis is always equal to the radius, which is 5 units.
• New y-coordinate = 5.

So, the new center of the circle after rolling is $(5 + 10\pi, 5)$.

The radius of the circle remains unchanged during rolling, so the new radius is still $r = 5$ units.

The equation of the circle in its new position, with center $(h', k') = (5 + 10\pi, 5)$ and radius $r' = 5$, is:

$(x - h')^2 + (y - k')^2 = (r')^2$

Substituting the new center coordinates and radius:

$(x - (5 + 10\pi))^2 + (y - 5)^2 = 5^2$

This equation represents the circle's position after one complete roll on the positive x-axis.

Comparing this with the given options, we find that it matches option 3.

Revision Table: Circle Rolling on Axis

Additional Information: Rolling Circle Problems

Understanding how shapes move is key in geometry. When a circle rolls without slipping along a line, the distance covered is directly related to its rotation.

• Rolling on a Straight Line: The center of the circle moves parallel to the line at a constant height equal to the radius. The horizontal distance covered by the center is equal to the arc length of the circle that has touched the line. For one complete roll, this distance is the full circumference.
• Rolling on a Curve: If a circle rolls on a curved path (like another circle), the path traced by the center is more complex. The distance covered by the center is related to the arc length of the path it follows, not just the arc length of the circle itself.
• Parametric Equations: The path of a point on the circumference of a rolling circle is called a cycloid (if rolling on a line) or epicycloid/hypocycloid (if rolling on a circle). These paths are described using parametric equations involving the angle of rotation.
• Applications: Rolling motion is fundamental in physics and engineering, from wheels and gears to the movement of planets.

For problems like this, always first identify the initial center and radius. Then, determine the distance and direction of rolling. Finally, calculate the new coordinates of the center and use the standard circle equation form.