Complete the square for general quadratic conic equations; identify and graph circles, ellipses, parabolas, or hyperbolas.
Decide whether a conic equation is normal, degenerate, or has no real graph by inspecting squared terms and signs
Problem
Does \((x-1)^2+(y+2)^2=-4\) describe a normal conic, a degenerate conic, or no real graph?
Big Picture
What this problem is really about
The visible conic pattern is only the first check; the required squared measurements must also be possible over the reals. Use the fact that every real square is nonnegative, combine those restrictions, and compare the result with the equation's constant. That comparison distinguishes an ordinary locus, a collapsed boundary case, and an empty real locus without trying to sketch imaginary points.
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