Locus

Locus:

The Best way to learn about a Locus is to understand that a Locus is something that satisfies some condition,we will learn about this in detail,also we will see the example of a Locus.

Definition of locus:

A locus is the set of all points (usually forming a curve or surface) satisfying some condition, or having a common property.

For example, the locus of a set of points all equidistant from a fixed point forms a circle in a two dimensional plane and a sphere in a 3-dimensional space. This means that if we traced all the points lying at the same distance ( called radius) from a point ( called centre) on a paper ( a two-dimensional space), it will form a circle, as can be seen if we keep one point of a compass fixed( at the centre) and move the other point on the paper to form a circle.

Different rules will create different shapes. Many geometric object have alternate definitions using the concept of locus. For example:

* Straight line "The locus of all points equidistant from two given points".
* Ellipse "The locus of all points where the sum of the distance to two fixed points is a constant."


Equation of a locus:

Equation to a locus is the algebraic relation that exists between x and y coordinates of a general point on the locus.

Ex:1Describe the locus of points that are 6 units from the point (3,-1) and give the equation of the locus.

All the points at a fixed distance ( 6 units) from a fixed point( 3,-1) form a circle of radius 6 units and centre (3,-1).

To find the equation, simply put in the geometrical formula and form an equation.

Square of the Distance of a point (x,y) from (3,-1) is given by ( 3 - x)2 + ( - 1 - y) 2

This should be equal to 6*6 or 62

So, the locus of all points (x,y) that are at a distance 6 units from (3,-1) follow the equation :

( 3 - x) 2 + ( - 1 - y)2 = 62


Example of locus of a point:


Three circles with the locus of a point of centers at A, B and C touch each other externally and AB = 4 cm BC = 6 cm CA = 8 cm. Find their radii.Solution:

There circles with the locus of a point of centers A, B, C touch one another. Let r1, r2, r3 be the radii of the circles.

Given:

AB = r1+ r2 = 4; BC = r2 + r3 = 6; CA = r3 + r1 = 8

Adding we get

2r1+ 2r2 + 2r3 = 18; 2 ( r1 + r2 + r3) = 18; r1 + r2 + r3 = 9

r1 = (r1+r2+r3) – (r2+r3) = 9–6 = 3

r2= (r1+r2+r3) – (r1+r3) = 9–8 = 1;

r3 = (r1+r2+r3) – (r1+r2) = 9–4 = 5

Radii are 3 cm, 1 cm and 5 cm respectively.

Hope you like the above example of Locus.Please leave your comments, if you have any doubts.