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Wednesday, 1 January 2014

Polar Co-ordinates

 

Beside the Cartesian , there are other systems also for  representing points and curve analytically . Polar system which is one of them .

---------In this system we started with a fixed line    OX   called the Initial Line and a fixed point on it , called the Pole .

---------If   P   be any given point, the distance   OP=r   is called the Radius Vector and    Angle XOP=theta    the Vectorial Angle . The two together are referred to as the Polar Co-ordinates of     .

Unrestricted Variation of Polar Co-ordinates :-

If we concerned with  assigning polar co-ordinates to only individual points in the plane , then it would clearly be enough to consider the radius vector to have positive values only and the vectorial angle    theta     to lie between    0 and 2pi   .

Transformation of Co-ordinates :-

Take the initial line   OX    of the polar system as the positive direction of   X-axis and the Pole   O    as origin for the Cartesian system .The positive direction of  Y-axis is to be such that the line   OX    after revolving through     Half Pi   is counter - clockwise direction comes to consider with it .

Polar Equation of Curves :-

Any explicit and implicit relation between   r    and   theta      will give a curve determined by the points whose co-ordinates satisfy that relation .

       ---------  Thus the Equations .

                    r=f(theta)       or       F(r,theta)    

                   determines  Curves .