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Wednesday, 19 February 2014

Orthogonal Trajectories

 


Definition :-  A curve which cuts every member of given family of curves according to a given law is called a trajectory of the given family .

----   We shall consider only the case when each trajectory cuts every member of a given family at a constant angle . The trajectory will be called orthogonal , if the constant angle is a right angle . For example , every line through the origin of co-ordinates is an orthogonal trajectory of the family of concentric circle with center at the origin .

How to fined the orthogonal trajectories of the family of curves  
  
f(x,y,c)=0

where    c    is a parameter .

Let ,  phai(x,y,dy/dx)=0 

be the differential equation of the family of curves given by

f(x,y,c)=0

 If     (dy/dx)=m   at a point   (x,y)  on one of the curves of the system and if another curve cuts that curve at right angle , then    m'   its slope must be given by the equation

mm'=-1

therefore ; m'=-1/m=-dx/dy 

Hence at  (x,y)  on the orthogonal trajectory , these equation must be satisfied

phai(x,y,dx/dy)=0

Hence this is the differential equation of the orthogonal system .