Consider the set $$S=\{x,y,z|2x^2+y^2+z^2+xz<5\}$$ how can we find the largest level set of $S$ on $x-y$ plain? basically I look for a compact set $\bar{S}(x,y)$ such that every $x,y\in S$ is also in $\bar{S}$. Is the following correct? the max of x when $y=0$ and $x,y,z\in S$ happens when $$4x+z=0$$ as a result we can conclude that every $x,y\in S$ also belongs to $$14x^2+y^2<5$$
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You asked a new question. The original question was about the projection of $S$ on the plane $xOy$. – user64494 Aug 17 '13 at 18:41
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I think I had chosen a wrong title but my question was exactly what I rephrased here (for more clarification). Thank you. – Eric Brown Aug 19 '13 at 23:04
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Your words do not correspond to reality: the body of the question was changed too. – user64494 Aug 20 '13 at 17:51
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The projection on the $XOY$-plane is defined by the equations $X=x,Y=y,Z=0.$ Therefore, the projection of $S$ on the plane $XOY$ is defined by the inequality $$2X^2+Y^2<5.$$ The commands of Maple $$with(plots): with(plottools):$$ $$P := implicitplot3d(2*x^2+x*z+y^2+z^2 < 5, x = -4 .. 4, y = -4 .. 4, z = -4 .. 4, numpoints = 10^4, style = surface); $$ $$ Q := project(P, [[0, 0, 0], [1, 0, 0], [0, 1, 0]]):display(Q);$$ confirm that.
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