Finding equations of simple loci

Understanding Loci

A locus (plural loci) is a set of points satisfying a certain condition. It can be a line, a curve, or any other shape.
The term “loci” is used primarily in mathematics and genetics.
In mathematics, the study of loci involves finding the path traced by a point that satisfies a given condition.

Finding Equations of Simple Loci

The process of finding equations for loci frequently involves using the definition of a geometric property. For example, a circle could be defined as the locus of points a constant distance (the radius) from a certain point (the centre).
The equation of a locus can typically be represented in a Cartesian form (x, y) or a parametric form, where coordinates of points on the locus are expressed as functions of a parameter.

Some Common Loci and Their Equations

The line: given two points, the locus of points equidistant from them forms a straight line. The equation is obtained using the slope-intercept form, y = mx + c where m is the slope and c is the y-intercept.
The circle: given a point, the locus of points at a constant distance (the radius) from that point forms a circle. This gives the standard form of a circle’s equation, (x-h)²+(y-k)²=r², where (h, k) are the center of the circle and r is the radius.
The parabola: this is the locus of points equidistant from a given point (the focus) and a given line (the directrix). This leads to the standard form of a parabola’s equation, y=a(x-h)²+k, where (h, k) is the vertex of the parabola.

Practical Applications of Loci

Understanding loci and their equations is important in a variety of subject areas, including physics, engineering, and computer graphics. It can support everything from understanding the path of a moving object to planning the layout of a building.
Familiarity with the concept of loci and how to find their equations is also useful in areas beyond geometry, such as calculus and complex numbers.

Remember, practice is key for understanding loci and being able to determine their equations. With consistent practice, you can become proficient in interpreting, defining, and deriving loci.