# Locus

• ### Prerequisites

1) Distance Formula

Distance between points ${(x_1, y_1)}$ and ${(x_2,y_2)}$ is given by

${d = \sqrt {(x_2-x_1)^2 + (y_2 - y_1)^2}}$

2) Section Formula

Let a point ${M (x,y)}$ divide the segment ${PQ}$ in the ratio ${m:n}$ internally. Internal division implies the order of points is ${P-M-Q}$. Let coordinates of points ${P}$ and ${Q}$ be ${(x_1,y_1)}$ and ${(x_2,y_2)}$ respectively. Then,

${\frac {l(PM)}{l(MQ)} = \frac {m}{n}}$

and

${M \equiv \Big (\frac {mx_2 + nx_1}{m+n} , \ \frac {my_2 + ny_1}{m+n} \Big )}$

If the point ${M}$ divides ${PQ}$ externally (order ${M-P-Q}$ or ${P-Q-M}$) in the ratio ${m:n}$, then,

${M \equiv \Big (\frac {mx_2 - nx_1}{m-n} , \ \frac {my_2 - ny_1}{m-n} \Big )}$

Midpoint Formula is obtained when the ratio ${m:n}$ is ${1:1}$.

3) Centroid of a Triangle whose vertices are ${(x_1,y_1)}$, ${(x_2,y_2)}$ and ${(x_3,y_3)}$ is

${(\bar x , \bar y) = \Big ( \frac {x_1 + x_2 + x_3}{3}, \frac {y_1 + y_2 + y_3}{3} \Big )}$

Knowing these 3 will be extremely useful to solve problems.

• ### Idea of Locus

We notice many geometrical shapes in our day-to-day lives. Wheels are circular, railway tracks are both straight and curved, the trajectory of a football kicked traces a particular curve, cross-section of an egg is oval, football field is rectangular the road-signs 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 set-builder form,

${L = \{ \ P | P \ satisfies \ a \ condition \} }$

• ### Equation of Locus

The points are specified using a coordinate system. Thus, each point is represented by an ordered pair ${(x,y) , \ s.t. \ x,y \in \mathbb R}$.

The equation of locus is a relationship between ${x}$ and ${y}$. 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 ${x}$ and ${y}$ in the equation of locus, ${LHS = RHS}$

If a point does not belong to a locus, ${LHS \ne RHS}$.

Different Curves as Loci

I) Line : Locus of points, under the condition ${\frac {y_2 - y_1}{x_2 - x_1} = constant}$, 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