Circles at the 700-800 Level

Almost every hard circle item begins the same way: the circle is handed to you in general form, and nothing can happen until you complete the square. The difficulty is never the completing — it is what gets asked afterwards.

Six archetypes cover the bank. General form to center and radius, including the variant where a leading coefficient must be divided out first. Where the circle meets an axis or a horizontal line, where the chord on $y = c$ has half-length $\sqrt{r^{2} - (c-k)^{2}}$ and tangency is the case where that radicand is zero. , which is perpendicular to the radius there. , turning a discriminant slog into one absolute-value equation. , where items deliberately mix degrees and radians in a single question. , whose area is and never .