Given a family of curves $F=(x-t)^2+y^2-\frac{1}{2}t^2,$ I am trying to compute the envelope of this family.
The envelope is described by the equations $F=0, \\ \dfrac{\partial F}{\partial t} = -2(x-t)-t = 0.$
To eliminate $t$, I computed a Gröbner basis for $I=\left\langle F,\dfrac{\partial F}{\partial t}\right\rangle.$ Namely, $ \{ g_1=x^2-y^2,g_2=t-2x \}.$ So $I\cap \mathbb R[x,y]= \langle g_1 \rangle$, and the envelope lies on the curve $x^2-y^2=0.$
Now, I want to draw a picture to illustrate my answer by Mathematica, (I also cannot imagine what the picture of that family looks like), but I don't know how to write the code. Could you give me a hand?