Problems from Reduced and canonical equations of the conics

Development:

To begin with, notice that there is no term $xy$, so the first reduction is not necessary because the main matrix $A^{\prime }$ is already diagonal.

Completing the squares for $x$, we see that the equation becomes $$(x+1{)}^2+y^2+2=0$$

Doing the change of variable $x^{\prime }=x+1,y^{\prime }=y$ we are left with the equation $$x^{\prime 2}+y^{\prime 2}+2=0$$ Notice that the canonical equation is that of an imaginary ellipse.

Solution:

The canonical equation is $x^{\prime 2}+y^{\prime 2}+2=0$ and therefore this is an imaginary ellipse.

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