Find the orthogonal trajectories of the family of curves: (x^2/a^2)+y^2/(a^2+w)=1

Answers (1)

Solution:  Given ,

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

             Diffrentiating (1) ,  w.r.t  x ,

            Rightarrow     frac2xa^2+frac2ya^2+wfracmathrmd ymathrmd x=0Rightarrow frac1a^2+w=-fracxa^2yfracmathrmd xmathrmd y   ......(2)

         Eliminating  w  fram (1) and (2) , we get

           Rightarrow   fracx^2a^2+y^2(-fracxa^2yfracmathrmd xmathrmd y)=1      or  fracx^2a^2-fracxya^2frac1fracmathrmd ymathrmd x=1,  ......(3)

       Which is the diffrential equation of the given family of curve (1).

       Replacing  fracmathrmd ymathrmd x   by -fracmathrmd xmathrmd y  in (3)  the differential equation of the orthiogonal trajectories is

      Rightarrow       fracx^2a^2-fracxya^2(-frac1fracmathrmd xmathrmd y)=1   or    fracx^2a^2-fracxya^2fracmathrmd ymathrmd x=1

     Rightarrow      fracxya^2fracmathrmd ymathrmd x=fraca^2-x^2a^2       or     ydy=[fraca^2x-x]dx

    Integrating ,     fracy^22=a^2log x-fracx^22+fracc2Rightarrow x^2+y^2-2a^2log x=c.

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