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. The tangent at a given point, which is perpendicular to the radius there. Tangency as a distance condition, turning a discriminant slog into one absolute-value equation. Sectors, arcs, and rotation, where items deliberately mix degrees and radians in a single question. The annulus, whose area is $\pi[(r+w)^{2} - r^{2}]$ and never $\pi w^{2}$.

The governing trap is $r^{2}$ wearing $r$'s clothes: completing the square leaves you holding $r^{2}$, and every question then wants something built from $r$. Write 'r =' on your scratch paper, not 'r squared ='.