Locus

A locus is a set of points, in geometry, which satisfies a given condition or situation for a shape or a figure. The plural of the locus is loci. The area of the loci is called region. The word locus is derived from the word location. Before the 20th century, the geometric shapes were considered as an entity or place where points can be located or can be moved. But in the modern Maths, the entities are considered as the set of points which satisfies the given condition.

The set of all points which forms geometrical shapes such as a line, a line segment, circle, a curve, etc. and whose location satisfies the conditions is locus. So, basically, we can say, instead of seeing them as a set of points, they can be seen as places where the point can be located or move.

Locus of Circle

With respect to the locus of the points or loci, the circle is defined as the set of all points equidistant from a fixed point, where the fixed point is the centre of the circle and the distance of the sets of points is from the centre is the radius of the circle. Let us say, P is the centre of the circle and r is the radius of the circle, from point p to the set of all points or to the locus of the points. See the picture below to understand the concept.

Locus Examples in Two Dimensional Geometry

We have already discussed the locus of the points which defines the path for a circle. Now let us see some more examples in 2-D geometry or plane geometry.

Perpendicular Bisector

The set of points which bisects the line, formed by joining two points and are equidistant from two points, is called as perpendicular bisector.

Angle Bisector

A locus or set of points which bisects an angle and are equidistant from two intersecting lines, which forms an angle, is called angle bisector.

Ellipse

The sets of points which satisfies the condition where the sum of the distances of two focus point is a constant defines an ellipse.

Locus of Ellipse

Parabola

The set of points or loci, which are equidistant from a fixed point and a line, is called a parabola. The fixed point is the focus and the line is the directrix of the parabola.

Locus of Parabola

Hyperbola

A hyperbola has two focus points, which are equidistant from the centre of the semi-major axis. The set of points, which satisfies the condition where the absolute value of the difference between the distances to two given foci is a constant defines a hyperbola.

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