Circles — 700-800

Circles: The 700-800 Patterns

Part 1 of 3 — The Archetypes Hard-Tier Items Are Built From

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.

Archetype 1: General Form → Center and Radius

$x^{2} + y^{2} + Dx + Ey + F = 0$

Halve each linear coefficient, square it, add it to both sides. From $x^{2} + y^{2} - 6x + 4y - 12 = 0$:

$(x - 3)^{2} + (y + 2)^{2} = 12 + 9 + 4 = 25$

Center $(3, -2)$, radius $5$ — not $25$. The single most reliable trap in this entire topic is an option built from $r^{2}$ where $r$ belongs.

The leading-coefficient variant. If the equation opens $2x^{2} + 2y^{2} - \ldots$ or $3x^{2} + 3y^{2} - \ldots$, you must divide through first. Completing the square on the undivided equation produces a clean, plausible, and completely wrong radius.

Archetype 2: Where the Circle Meets an Axis or a Horizontal Line

To find where a circle crosses the $y$-axis, set $x = 0$ and solve the resulting quadratic in $y$; the chord length is the distance between the two roots. A line $y = c$ works identically. Two facts do the work:

• The chord on $y = c$ has half-length $\sqrt{r^{2} - (c - k)^{2}}$, where $(h, k)$ is the center.
• If that expression is zero, the line is tangent — which is how "meets at exactly one point" items are built.

Archetype 3: Tangent Line at a Given Point

The tangent is perpendicular to the radius at the point of contact. So: slope of radius $\rightarrow$ negative reciprocal $\rightarrow$ point-slope through the contact point $\rightarrow$ whatever intercept was requested. Using the radius's own slope is the planted error.

Archetype 4: Tangency With No Contact Point Given

When a line is tangent to a circle but the contact point is not given, tangency means exactly one intersection point.

• Horizontal or vertical line: $y = c$ is tangent exactly when $|c - k| = r$, and $x = c$ exactly when $|c - h| = r$. "Tangent to the $x$-axis" means $r = |k|$.
• Slanted line such as $y = x + b$: substitute it into the circle's equation, collect into one quadratic in $x$, and set the discriminant equal to $0$. Solve for the parameter, then check the condition in the stem ("the positive value") to choose between the two answers.

Archetype 5: Sectors, Arcs, and Rotation

Two formulas, and the units decide which:

• Radians: arc $= r\theta$, sector area $= \frac{1}{2}r^{2}\theta$.
• Degrees: arc $= \frac{\theta}{360}(2\pi r)$, sector area $= \frac{\theta}{360}(\pi r^{2})$.

Hard items deliberately put one angle in degrees and the other in radians in the same question. Rolling-wheel and revolution problems are arc length wearing a costume: one revolution is $2\pi$ radians, and rolling without slipping means distance travelled $=$ arc length.

Archetype 6: The Annulus

A path of uniform width $w$ around a circle of radius $r$ has area $\pi\left[(r + w)^{2} - r^{2}\right]$ — never $\pi w^{2}$. Multiply by a depth for volume, then by a unit price for cost, and check which of those three the question actually wants.

Circles: Traps & Speed

Part 2 of 3 — Distractor Autopsy

Distractor Species 1: $r^{2}$ Wearing $r$'s Clothes

Completing the square leaves you holding $r^{2}$. Every question then wants something built from $r$ — a circumference, a distance, a chord. The option computed from $r^{2}$ is present in almost every item in this bank. Write "$r =$" on your scratch paper, not "$r^{2} =$."

Distractor Species 2: Distance to the Center vs. Distance to the Circle

"How far is $P$ from the circle?" is not "how far is $P$ from the center." For an external point:

$\text{shortest distance to the circle} = PC - r$

and the distance to the far side is $PC + r$. Both $PC$ and $PC + r$ are always offered.

Distractor Species 3: The Sign Slip Reading the Center

$(x + 3)^{2}$ means the center's $x$-coordinate is $-3$, not $3$. A center misread this way still produces a perfectly clean answer, which is why it survives all the way to the bubble.

Distractor Species 4: Same Side vs. Opposite Sides

For two parallel chords, compute each chord's distance from the center. If the chords are on the same side, subtract those distances; on opposite sides, add them. The unwanted one is always an option, and the question states which case you are in — usually in a single clause you are moving too fast to read.

Distractor Species 5: Degrees Where Radians Belong

$\frac{5\pi}{6}$ is an angle in radians ($150^{\circ}$). Dropping it into a degree formula, or dropping $135^{\circ}$ into $\frac{1}{2}r^{2}\theta$, both produce wrong answers that look reasonable. Convert everything to one system before touching a formula.

Distractor Species 6: Stopping at the Wrong Layer

Sprinkler, gravel, and tile items run area $\rightarrow$ volume $\rightarrow$ cost. Each layer is an option. The question's final noun tells you where to stop: square meters, cubic meters, liters, or dollars.

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Speed Techniques

Complete the square in one pass. For $x^{2} + y^{2} + Dx + Ey + F = 0$, the center is $\left(-\frac{D}{2}, -\frac{E}{2}\right)$ and $r^{2} = \frac{D^{2}}{4} + \frac{E^{2}}{4} - F$. Reading the center straight off the coefficients saves twenty seconds and removes the sign slip entirely.

Chord on a horizontal line, instantly. For center $(h, k)$ and radius $r$, the line $y = c$ cuts a chord of length $2\sqrt{r^{2} - (c - k)^{2}}$. Tangency is exactly the case where that radicand is zero.

Tangency, fastest route first. For a horizontal or vertical line, compare the center's coordinate with $r$: $y = c$ is tangent when $|c - k| = r$. For a slanted line, substitute and set the discriminant to $0$. When the contact point is given, use the fact that the radius is perpendicular to the tangent.

Sector shortcut. In radians, the sector area is $\frac{1}{2}r^{2}\theta$ and the arc is $r\theta$ — so $\text{area} = \frac{1}{2} \cdot r \cdot \text{arc}$. If a question gives you the sector area and asks for the arc, that relationship gets there in one step.

Circles: Timed Drill

Part 3 of 3 — Four Items at Test Pace

About 90 seconds each. These four cover the highest-frequency hard-tier skeletons: parallel chords, rolling-wheel arc length, the annulus cost chain, and the chord-plus-center triangle.

Run the checklist before every answer:

1. Is my $r$ actually $r$, or is it still $r^{2}$?
2. Radians or degrees — and did the question mix them?
3. Which layer was requested? Length, area, volume, or dollars.
4. Same side or opposite sides? Subtract or add.