Find the canonical equation of the conic defined by the following equation $$x^2+y^2+2x+3=0$$.
See development and solution
Development:
To begin with, notice that there is no term $$xy$$, so the first reduction is not necessary because the main matrix $$A'$$ 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 $$x' = x+1, \ y' = y$$ we are left with the equation $$$x'^2+y'^2+2=0$$$ Notice that the canonical equation is that of an imaginary ellipse.
Solution:
The canonical equation is $$x'^2+y'^2+2=0$$ and therefore this is an imaginary ellipse.