The locus of a point moving on the arc of a circle

Understanding Locus on the Arc of a Circle

A locus is defined as a set of points satisfying a particular condition or a set of conditions.
In geometry, when we talk about the locus of a point moving along the arc of a circle, we refer to the set of all points that a given point can reach while moving along that arc.
The common condition here is that the point remains an invariant distance from the center of the circle- equal to the radius of the circle.

Key Features of a Circle

A circle is defined by two components, its center and its radius. The center is a fixed point, and the radius is the constant distance from the center to the boundary of the circle.
An arc of a circle is a contiguous part of the circle’s circumference. The locus of a point moving on an arc would be a smaller subset of the full circle.

Values and Properties

When a point is moving along an arc, it is important to remember that, at any position on the arc, the line drawn from the point to the centre of the circle creates an angle at the center.
This central angle is variable as the point moves along the arc, but the radius (or the line connecting the point to the center) always remains constant.
A complete rotation along the arc of a circle equates to an angular measurement of 360 degrees or 2π radians.

Relevance to Coordinate Geometry

The concept of a locus helps in determining and visualising a path or a set of points in a plane.
As such, the locus of a point moving on the arc of a circle plays a fundamental role in understanding circle geometry and coordinate geometry.
This concept is also pivotal in learning advanced topics such as Complex Numbers and their geometric interpretation.

Ultimately, the key to mastery of this topic is to carefully study the definitions, properties and applications of circles, arcs, and locuses. Regular practice with example problems and diagrams is crucial for understanding and application of the concept. Using reference books, online educational resources and attempting past papers are recommended for reinforcement of these concepts.