Changing the variables

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We define a new coordinate system $R = x + y,$ $S = x - y,$ in which $R$ and $S$ are tilted at 45 degrees relative to $x$ and $y.$ This gives us

  $\displaystyle x $ $\displaystyle = $ $\displaystyle \frac{1}{2}(R+S) $    
  $\displaystyle y $ $\displaystyle = $ $\displaystyle \frac{1}{2}(R-S). $    

Substituting this into our expression for the envelope curve

  \[ y(x)=(1-\sqrt{x})^2 \]    

gives

  $\displaystyle \frac{1}{2}(R-S) $ $\displaystyle = $ $\displaystyle \left(1-\sqrt{\frac{1}{2}(R+S)}\right)^2 $    
  $\displaystyle \frac{1}{2}(R-S) $ $\displaystyle = $ $\displaystyle 1+\frac{1}{2}(R+S) - 2\sqrt{\frac{1}{2}(R+S)} $    
  $\displaystyle 0 $ $\displaystyle = $ $\displaystyle 1+S -\sqrt{2(R+S)} $    
  $\displaystyle 2R+2S $ $\displaystyle = $ $\displaystyle 1+S^2+2S $    
  $\displaystyle 2R $ $\displaystyle = $ $\displaystyle 1+S^2 $    
  $\displaystyle $R $ $ $\displaystyle = $ $\displaystyle \frac{1}{2}+\frac{S^2}{2}. $    

This is the equation of a parabola.

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