Pedal Equation

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 ,

f(r,theta)=0  --------------------(1)

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 .