### 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

*at a point*

**(dy/dx)=m***on one of the curves of the system and if another curve cuts that curve at right angle , then*

**(x,y)***its slope must be given by the equation*

**m'**

**mm'=-1**therefore ;

**m'=-1/m=-dx/dy**Hence at

*on the orthogonal trajectory , these equation must be satisfied*

**(x,y)**

**phai(x,y,dx/dy)=0**Hence this is the differential equation of the orthogonal system .

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