Question

What is a locus? Explain elaborately

Answer 1

Each curve in this example is a locus defined as the conchoid of the point P and the line l. In this example, P is 8 cm from l.
In geometry, a locus (plural: loci) (Latin word for "place", "location") is a set of all points (commonly, a line, a line segment, a curve or a surface), whose location satisfies or is determined by one or more specified conditions.[1][2]
In other words, the set of the points that satisfy some property is often called the locus of a point satisfying this property. The use of singular in this formulation is a witness that, until the end of 19th century, mathematicians did not consider infinite sets. Instead of viewing lines and curves as sets of points, they viewed them as places where a point may be located or may move.


Examples from plane geometry include:
The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points.[8]
The set of points equidistant from two lines that cross is the angle bisector.
All conic sections are loci:[9]
Parabola: the set of points equidistant from a fixed point (the focus) and a line (the directrix).
Circle: the set of points for which the distance from a single point is constant (the radius). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is, not 1) is referred to as a circle of Apollonius.
Hyperbola: the set of points for each of which the absolute value of the difference between the distances to two given foci is a constant.
Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant. The circle is the special case, in which the two foci coincide with each other.
Other examples of loci appear in various areas of mathematics. For example, in complex dynamics, the Mandelbrot set is a subset of the complex plane that may be characterized as the connectedness locus of a family of polynomial maps.
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