What is Polar Co-ordinates?

**Besides Cartesian co-ordinate system we have several other methods for locating position of a point on a plane. Of all these system we shall make here a brief discussion on Polar Co-ordinates only. Polar Co-ordinates are widely used in higher mathematics as well as in other branches of science. **

In polar co-ordinate system the position of a point on the reference plane is uniquely determined referred to a fixed point on the plane and a half line drawn through the fixed point. The fixed point is called the *Pole* or *Origin* and the half line drawn through the pole is called the *Initial Line*.

Let OX be the initial line drawn through the pole O on the plane of reference. Take any point P on the plane and join OP .

If OP = r and ∠XOP = θ then the real numbers r and θ are together called the Polar Co-ordinates of P and denoted by (r, θ); here OP. If OP = r and *Polar Co-ordinates of P* and denoted by (r, θ); here OP = r is called the ** Radius Vector** and ∠XOP = θ, the

By convection, to represent the polar co-ordinate of a point we first write the radius vector (r) and then the vectorial angle (θ) and they are put together in braces putting a comma between them.

**Note:**

(i) for given values of r and θ we shall get one and only one point on the reference plane; conversely, for a given point on the plane r possesses a definite finite value but θ can have infinite number of value (viz ., θ, 2π + θ, 4π + θ,…….etc.).

(ii) The polar Co-ordinates of the pole are assumed to be (0, 0).

(iii) If the sense of radius vector is taken into account then the value of r may be negative. Thus, if the direction from O to P is taken as positive then the direction from P to O will be negative. Hence, if the points P, O, P’ are collinear such that OP = OP’ = r and ∠XOP = θ then the polar co-ordinates of P and P' are (r, θ) and (-r, θ) respectively.

However, in practice, it is convenient to take both the radius vector (r) and the vectorial angle (θ) as positive.

(iv) Remembering the rules regarding the signs of r and θ we can represent the polar co-ordinate of P in following different ways:

(r, θ); (-r, π + θ); [r, -(2π - θ)]; [-r, -(π - θ)].

**●**** Co-ordinate Geometry**

**What is Co-ordinate Geometry?****Rectangular Cartesian Co-ordinates****Polar Co-ordinates****Relation between Cartesian and Polar Co-Ordinates****Distance between Two given Points****Distance between Two Points in Polar Co-ordinates****Division of Line Segment****: Internal & External****Area of the Triangle Formed by Three co-ordinate Points****Condition of Collinearity of Three Points****Medians of a Triangle are Concurrent****Apollonius' Theorem****Quadrilateral form a Parallelogram****Problems on Distance Between Two Points****Area of a Triangle Given 3 Points****Worksheet on Quadrants****Worksheet on Rectangular – Polar Conversion****Worksheet on Line-Segment Joining the Points****Worksheet on Distance Between Two Points****Worksheet on Distance Between the Polar Co-ordinates****Worksheet on Finding Mid-Point****Worksheet on Division of Line-Segment****Worksheet on Centroid of a Triangle****Worksheet on Area of Co-ordinate Triangle****Worksheet on Collinear Triangle****Worksheet on Area of Polygon****Worksheet on Cartesian Triangle**

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