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Circles — 700-800

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Circles — 700-800 - Complete Interactive Lesson

Part 1: The 700-800 Patterns

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 . 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 is the center.

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 as a Distance Condition

When a line is tangent to a circle but no contact point is given, tangency means the distance from the center to the line equals $r$ :

$\frac{|Ah + Bk - C|}{\sqrt{A^{2} + B^{2}}} = r$

This turns "for what $k$ is $y = 2x + k$ tangent?" into a one-line absolute-value equation instead of a discriminant slog.

Archetype 5: Sectors, Arcs, and Rotation

Two formulas, and the units decide which:

Radians: arc $= r\theta$ , sector area $= \frac{1}{2}r^{2}\theta$ .

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 $\pi\left[(r + w)^{2} - r^{2}\right]$ — . Multiply by a depth for volume, then by a unit price for cost, and check which of those three the question actually wants.

Part 2: Traps & Speed

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 — a circumference, a distance, a chord. The option computed from is present in almost every item in this bank.

Part 3: Timed Drill

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:

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

divide through first

If that expression is zero, the line is tangent — which is how "meets at exactly one point" items are built.

Degrees: arc

= \frac{\theta}{360}(2\pi r)

, sector area

= \frac{\theta}{360}(\pi r^{2})

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:

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

and the distance to the far side is $\overline{PC} + r$ . Both $\overline{PC}$ and $\overline{PC} + r$ 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.

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 without a discriminant. Set the perpendicular distance from the center to the line equal to $r$ and solve the absolute-value equation. One line replaces a page of substitution.

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.