Changing the variables

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Changing the variables


We define a new coordinate system R=x+y, S=xy, in which R and S are tilted at 45 degrees relative to x and y. This gives us x=12(R+S)y=12(RS). Substituting this into our expression for the envelope curve y(x)=(1x)2 gives 12(RS)=(112(R+S))212(RS)=1+12(R+S)212(R+S)0=1+S2(R+S)2R+2S=1+S2+2S2R=1+S2$R=12+S22. This is the equation of a parabola.

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