From the general equation of a circle or its center and radius — get the center, radius, standard and general form, area and circumference, drawn to scale.
Give the analyzer a circle in general form or as a center and radius, and it rebuilds every other form of the equation, with the working shown.
A circle in general form looks like x² + y² + Dx + Ey + F = 0. The analyzer collects the x terms (x² + Dx) and the y terms (y² + Ey) and moves the constant F to the other side, ready for completing the square.
Adding (D/2)² and (E/2)² to both sides turns each group into a perfect square: (x + D/2)² + (y + E/2)² = D²/4 + E²/4 − F. Read off the center as (−D/2, −E/2) and the radius as the square root of the right-hand side.
If D²/4 + E²/4 − F is positive you have a real circle; if it is zero the equation describes a single point, and if it is negative there is no real graph at all. For a real circle the analyzer reports the area πr² and the circumference 2πr and draws it to scale on equal axes, so it looks round rather than stretched.