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Sunday, 9 March 2014

Polar Equations 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


or,  F=(r, theta)=0    determine curves ,

The co-ordinates of two points symmetrically situated about the initial line are of the form 

(r, theta)   and  (r, -theta)

So, that their vertical angles differ in sign only .

Hence a curve will be symmetrical about the initial line if ,

on changing   theta    to    -theta .

its equation does not change.  For instance the curve

r=a(1+cos theta)

is symmetrical about the initial line , for ;

r=a(1+cos theta)=a[1+cos(-theta)]

It may be noted that ;

r=a  represents a circle with its center at the pole and radius  a  

theta=b   represents the line through the pole obtained by revolving the initial line through the angle   b.