The locus of a point moving in a circle

Understanding the Locus

The locus of a point refers to the path that this point follows according to a certain set of conditions or rules.
In the context of circles, the locus of a point moving in a circle is simply the circle itself.

The circle can be defined as the locus of all points that lie at a fixed distance from a given point, which we call the centre of the circle.
This fixed distance is known as the radius of the circle.
In other words, if a point is moving such that it always remains the same distance from the centre, it is moving in a circle.

The equation of a circle with centre at (h, k) and radius r is given by (x - h)² + (y - k)² = r².
If a point (x, y) satisfies this equation, it lies on the locus—that is, the circle.
If it doesn’t satisfy the equation, the point isn’t on the locus and therefore isn’t on the circle.

Geometrically, the circle is a perfect symmetry. No matter the orientation, the shape remains the same.
The diameter is the longest line which can be drawn within the circle, passing through the centre. It is twice the radius.
The circumference of a circle is the path or the locus that the point takes. It can be calculated using the formula 2πr, where r is the radius of the circle.

Visualisation can significantly aid in understanding the concept of a locus. Draw diagrams to help visualise the locus of a point moving in a circle.
When finding the locus in more complex situations, remember that it’s essentially about satisfying certain conditions, such as maintaining a fixed distance (for a circle).
Frequently reviewing and using the equation of a circle is a good way to reinforce your understanding of circular loci.
Considering the geometric properties of circles helps to cement understanding of this type of loci—practice can make perfect!