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# Locus and Polar Coordinates

Locus for a Point can be defined as a curve or a path that results from the condition (s) that is satisfied by the point for an Algebraic Relation governed by some fixed rule. This statement is valid only for those points which are lying in the plane and not in the outer region of the plane, i.e., the equation of loci cannot be found for those points lying in the exterior.

And as we know that a point can be located or represented in a plane using two forms of systems namely the Rectangular/Cartesian Coordinate System or the Polar Coordinate System.

A proper relation can be framed between the locus and the Polar Coordinates by using these coordinates to find the locus of any arbitrary point lying in the plane.
The simple technique which can be followed for doing this is by substituting the values of polar coordinates in place of x and y, i.e.,

x = h cos A
y = h sin A

The steps for finding the equation of the locus of a point P being the same:
First of all assign the coordinates to the point of which the locus has to be found, i.e. P (n, m).
The second thing which we need to remember is to express the given conditions as equations in terms of the known quantities and unknown parameters.
The unknown parameters should be eliminated, such that only the know quantities are left for consideration.
Replace n by x, and m by y. The resulting equation representing the locus of the point P can be obtained by replacing n by x, and m by y.

All the points lying in the plane and making the same angle with the positive x- axis will have the same Slope and also the equation of the loci.