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

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