Find the orthogonal trajectories of the family of the curves x^2/a^2+y^2/a^2+w=1 where w is parameter

Answers (1)

Solution:  We have ,

                        fracx^2a^2+fracy^2a^2+w=1           ........(1)

                Diffrentiating (1) with respect to x , we have

         Rightarrow                      frac2xa^2+frac2ya^2fracmathrmd ymathrmd x=0        ........(2)

                                    x+yfracmathrmd ymathrmd x=0

           Rightarrow     Now , to  obtain orthogonal trajectory replace fracmathrmd ymathrmd x by -fracmathrmd xmathrmd y

        Rightarrow         x+y(-fracmathrmd xmathrmd y)=0Rightarrow fracdyy=fracdxx  

         Integrating both sides we get    ln y=ln x+ln cRightarrow y=xc   

       

        

 

                

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