Question

Curve generated by a point on the circumference of a circle which rolls along a straight line is termed:

• A. Hypocycloid
• B. Epicycloid
• C. Cycloid
• D. Trochoid

Hint:

In physics and engineering, cycloids are used to model phenomena like the motion of wheels and the design of gear teeth.

Answer 1

The Correct Option is C

Step 1: Understanding cycloid.
A cycloid is the curve traced by a point on the circumference of a circle as the circle rolls along a straight line. This curve is commonly used in mechanical applications such as gear teeth.

Step 2: Analyzing the options.
(A) Hypocycloid: A hypocycloid is the curve traced by a point on the circumference of a circle rolling inside another circle, not along a straight line.
(B) Epicycloid: An epicycloid is the curve traced by a point on the circumference of a circle rolling outside another circle, not along a straight line.
(C) Cycloid: Correct — A cycloid is the curve generated by a point on the circumference of a circle rolling along a straight line.
(D) Trochoid: A trochoid is a broader class of curves that includes both cycloids and other types of curves traced by points on circles rolling along a straight line.

Step 3: Conclusion.
The correct answer is (C) Cycloid, as it is the curve generated by a point on the circumference of a circle rolling along a straight line.

Answer 2

Each of these curves is defined by exactly how a circle rolls, so matching the described motion, rolling along a straight line, to the correct definition solves this directly.

1. Hypocycloid: Traced by a point on a circle that rolls inside another, larger circle, an entirely different rolling surface from a straight line.
2. Epicycloid: Traced by a point on a circle that rolls outside another, fixed circle, again a curved rolling surface, not a straight line.
3. Cycloid: Traced by a point on a circle that rolls along a flat, straight line, matching the exact motion described in the question.
4. Trochoid: A broader family of curves that includes cycloids as well as curves traced by points inside or outside the rolling circle (not just on its circumference), so it is a more general category rather than the specific curve asked about.

Since the question specifies rolling along a straight line with the point exactly on the circle's edge, this is the defining case of the cycloid specifically, not the broader trochoid family or the circle-on-circle curves.

Therefore, the correct answer is Cycloid.