## 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.

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

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 6^2

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 = 6^2

**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 6^2

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 = 6^2