
Prerequisites
1) Distance Formula
Distance between points and is given by
2) Section Formula
Let a point divide the segment in the ratio internally. Internal division implies the order of points is . Let coordinates of points and be and respectively. Then,
and
If the point divides externally (order or ) in the ratio , then,
Midpoint Formula is obtained when the ratio is .
3) Centroid of a Triangle whose vertices are , and is
Knowing these 3 will be extremely useful to solve problems.

Idea of Locus
We notice many geometrical shapes in our daytoday lives. Wheels are circular, railway tracks are both straight and curved, the trajectory of a football kicked traces a particular curve, crosssection of an egg is oval, football field is rectangular the roadsigns are written on triangular boards and so on.
If these curves are drawn on a paper and analyzed, it can be seen that, all points belonging to a particular curve satisfy certain conditions. For example, points on a circle are equidistant from its center and the distance is known as the radius of the circle.
Informally, we have defined the locus. It is a Latin word with plural loci.

Definition
Locus is a set of points, satisfying certain (geometrical) condition. Using the setbuilder form,

Equation of Locus
The points are specified using a coordinate system. Thus, each point is represented by an ordered pair .
The equation of locus is a relationship between and . This relationship is an algebraic interpretation of the locus and hence is known as the equation of locus.
All points belonging to a locus satisfy its equation. In other words, by substituting and in the equation of locus,
If a point does not belong to a locus, .
Different Curves as Loci
I) Line : Locus of points, under the condition , for all pairs of points
II) Circle : Locus of points in a plane, which are equidistant from a fixed point
III) Ellipse : Locus of points in a plane, such that sum of distances of any point from 2 fixed points is constant
IV) Hyperbola : Locus of points in a plane, such that difference of distances of any point from 2 fixed points is constant
V) Parabola : Locus of points in a plane, such that distance of any point from a fixed point is equal to the distance of the same point from a fixed line
VI) Cycloid : Locus of a point on the edge of circle, which rolls without slipping