Testing Whether Equations Represent Circles
The discriminant test using D²/4 + E²/4 - F is your best friend for determining what an equation represents. Let's test three different scenarios.
For x² + y² + 12y = 0: D=0, E=12, F=0. The test gives 0 + 36 - 0 = 36 > 0, so it's a circle. For x² + y² - 4x + 2y = -5: D=-4, E=2, F=5. The test gives 4 + 1 - 5 = 0, so it's a point circle.
For x² + y² + 4x - 6y + 18 = 0: D=4, E=-6, F=18. The test gives 4 + 9 - 18 = -5 < 0, so it has no graph. This systematic approach prevents you from wasting time trying to graph impossible equations.
💡 Quick Check: If your discriminant is negative, don't even try to complete the square - there's no real circle there!