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,
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 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.
Locus is a set of points, satisfying certain (geometrical) condition. Using the set-builder 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