The (p , r) or Pedal Equation of Curve :-
We have been acquainted with two types of equation of any curve ; one Cartesian Equation (x,y) and the other the Polar Equation containing (r,theta) . When the equation of any curve is given in terms of (p,r) where p is the length of the perpendicular from the pole on the tangent and r is the radius vector , then that form of the curve is called the Pedal Equation .
Find the pedal equation of a curve from its polar form :-
Let the polar equation of any curve be ,
Let the coordinates of any point on the curve be (r,theta) and let the length of perpendicular from the pole on the tangent at (r,theta) be p .
If phai be the angle between the tangent and the radius vector ,
then we know that ,
tan (phai) = r .d theta /dr ----------(2)
and p=r . sin(phai) ---------------(3)
Now , if we eliminate theta between the equations (1) , (2) and (3) then we shall get an equation in terms of p and r and thus will be required an equation of the curve .