For parametric equation, where #f(x(t),y(t))#, #(dy)/(dx)# is given by #((dy)/(dt))/((dx)/(dt))#
Here #x(t)=a(t^2-1/t^2+1)^(1/2)#
therefore #(dx)/(dt)=a/(2(t^2-1/t^2+1)^(1/2))xx(2t+2/t^3)#
and as #y(t)=at(t^2-1/t^2+1)^(1/2)#
#(dy)/(dt)=a(t^2-1/t^2+1)^(1/2)+at(2t+2/t^3)/(2(t^2-1/t^2+1)^(1/2))#
Hence #(dy)/(dx)=(a(t^2-1/t^2+1)^(1/2)+at(2t+2/t^3)/(2(t^2-1/t^2+1)^(1/2)))/((a(2t+2/t^3))/(2(t^2-1/t^2+1)^(1/2)))#
= #(2a(t^2-1/t^2+1)+at(2t+2/t^3))/(a(2t+2/t^3))#
= #(2at(t^4+t^2-1)+2at(t^4+1))/(2a(t^4+1))#
= #(t(t^4+t^2-1+t^4+1))/(t^4+1)#
= #(t^3(2t^2+1))/(t^4+1)#
