GELE Mathematics — Analytic GeometryMemory Anchors

Tags

• formula
• distance
• Pythagorean theorem
• coordinate geometry

Topic

Points and Lines

Concept

Distance Formula: d = √[(x₂−x₁)² + (y₂−y₁)²]

Anchor Id

A1

Difficulty

easy

Memory Aid

Imagine a construction surveyor in Manila who needs to measure the diagonal distance across a rectangular lot. He can't walk diagonally, so he walks the horizontal leg (Δx) then the vertical leg (Δy). His total diagonal is the hypotenuse — Pythagorean theorem! The distance formula IS the Pythagorean theorem in disguise. Δx² + Δy² = d². Always square the legs, add them, then take the square root.

Anchor Type

micro_story

Why It Works

Connecting the abstract formula to the familiar Pythagorean theorem creates a conceptual bridge. The surveyor story adds emotional context and a professional engineering scenario relevant to Filipino CE reviewees.

Example Usage

Find distance from (1,2) to (4,6): Δx=3, Δy=4 → d=√(9+16)=√25=5. The surveyor walks 3 m east and 4 m north — diagonal is 5 m. Classic 3-4-5 right triangle!

Recall Trigger

Think: 'surveyor, right triangle, Pythagorean theorem'

Tags

• formula
• midpoint
• average
• coordinates

Topic

Points and Lines

Concept

Midpoint Formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)

Anchor Id

A2

Difficulty

easy

Memory Aid

The midpoint is like splitting the bill equally at Jollibee with one friend. You each pay HALF of the total. Midpoint = AVERAGE of x-coordinates and AVERAGE of y-coordinates. Add both x's and split in half. Add both y's and split in half. Always just the arithmetic mean — no squaring, no square roots, just average.

Anchor Type

analogy

Why It Works

The Jollibee reference is instantly relatable to Filipino students. 'Average' is a concept everyone understands. This eliminates the common mistake of trying to use square roots in the midpoint formula.

Example Usage

Midpoint of (1,2) and (4,6): x_mid=(1+4)/2=2.5, y_mid=(2+6)/2=4. Answer: (2.5, 4).

Recall Trigger

Think: 'split the bill equally — average of each coordinate'

Tags

• formula
• slope
• rise over run
• gradient

Topic

Points and Lines

Concept

Slope Formula: m = (y₂−y₁)/(x₂−x₁) — Rise over Run

Anchor Id

A3

Difficulty

easy

Memory Aid

RISE over RUN, never the other way — RISE is vertical (y up or down), RUN is horizontal (x left or right). Chant: 'Y on top, X below — RISE over RUN is how slopes go!' Remember: Y comes BEFORE X in the alphabet, and the numerator comes BEFORE the denominator — so Y is on top!

Anchor Type

rhyme

Why It Works

Rhymes exploit the phonological loop in working memory. The alphabetical trick (Y before X → numerator before denominator) provides a logical anchor that survives exam stress.

Example Usage

Slope from (1,2) to (4,6): m = (6−2)/(4−1) = 4/3. Rise=4, Run=3. For every 3 m horizontal, the grade rises 4 m.

Recall Trigger

Chant: 'Y on top, X below — Rise over Run!'

Tags

• formula
• perpendicular
• slope
• negative reciprocal

Topic

Points and Lines

Concept

Perpendicular Lines: m₁ × m₂ = −1 (negative reciprocal)

Anchor Id

A4

Difficulty

medium

Memory Aid

Use 'FLIP AND NEGATE' — to get the perpendicular slope, FLIP the fraction and NEGATE the sign. Example: if m=2/3, perpendicular slope is −3/2. The product always equals −1 because flipping and negating cancel each other out mathematically. Think of it as a 90° turn: you must completely change direction (negate) AND exchange rise and run (flip).

Anchor Type

mnemonic

Why It Works

The two-action phrase 'FLIP AND NEGATE' gives a procedure, not just a fact. Procedures are easier to recall under pressure than abstract rules.

Example Usage

Line 1 has slope 3/4. Perpendicular slope = −4/3. Check: (3/4)(−4/3) = −1 ✓. Common board trap: m=2 → perpendicular is −1/2, not −2.

Recall Trigger

Think: 'perpendicular = FLIP AND NEGATE the slope'

Tags

• formula
• point-to-line distance
• absolute value
• key formula

Topic

Points and Lines

Concept

Distance from a Point to a Line: d = |Ax₀ + By₀ + C| / √(A²+B²)

Anchor Id

A5

Difficulty

medium

Memory Aid

ABSO-LUTELY PLUG AND DIVIDE! A foreman needs to measure how far a pile (point P) stands from a property boundary line (Ax+By+C=0). Step 1: PLUG the pile's coordinates into the line equation — you get a number. Step 2: Take ABSOLUTE VALUE (distance is never negative!). Step 3: DIVIDE by the square root of A-squared plus B-squared (the 'normalizer'). The denominator is just √(A²+B²) — same as the length of the direction vector of the line.

Anchor Type

micro_story

Why It Works

The foreman-and-pile story creates a vivid professional image. The three-step procedure (PLUG → ABSOLUTE → DIVIDE) is an actionable sequence that prevents the common mistake of forgetting the absolute value.

Example Usage

Distance from (3,4) to 3x+4y−10=0: |3(3)+4(4)−10|/√(9+16) = |9+16−10|/5 = 15/5 = 3 units.

Recall Trigger

Think: 'foreman, pile, boundary — PLUG, ABSOLUTE, DIVIDE'

Tags

• formula
• parallel lines
• distance
• board exam favorite

Topic

Points and Lines

Concept

Distance between Two Parallel Lines: d = |C₁ − C₂| / √(A²+B²)

Anchor Id

A6

Difficulty

medium

Memory Aid

Two parallel train tracks (two parallel lines with the same A and B coefficients). The distance between them is the DIFFERENCE of their C values (how far each track is from the origin), divided by the same normalizer √(A²+B²). Think: the tracks run in the same direction (same A,B) but are offset (different C). The 'gap' is |C₁−C₂|/√(A²+B²).

Anchor Type

analogy

Why It Works

Train tracks are a perfect visual analogy for parallel lines. Filipino students are familiar with PNR or LRT tracks running parallel — the 'gap' between them is the distance formula in action.

Example Usage

Distance between 3x+4y−10=0 and 3x+4y+5=0: |−10−5|/√(9+16) = 15/5 = 3 units. Note: rewrite so both have SAME A and B before applying.

Recall Trigger

Think: 'parallel train tracks — gap = |C₁−C₂| / normalizer'

Tags

• formula
• circle
• conic
• center-radius form

Topic

Conic Sections — Circle

Concept

Circle Equation: (x−h)² + (y−k)² = r²

Anchor Id

A7

Difficulty

easy

Memory Aid

Visualize a MANHOLE COVER (circular, centered at a point on the street). The center of the manhole is at (h,k). Every point on the rim is exactly r meters away from the center. The equation literally says: 'squared horizontal distance from center' + 'squared vertical distance from center' = 'radius squared.' The signs inside the parentheses are MINUS h and MINUS k — because you're measuring distance FROM the center, so you subtract the center's coordinates.

Anchor Type

visual_association

Why It Works

The manhole cover is a tangible, Filipino-street-scene visual. Emphasizing the MINUS sign addresses the most common board exam trap where students write +h and +k instead of subtracting.

Example Usage

Circle center (2,−3), radius 5: (x−2)²+(y+3)²=25. Note: y−(−3) = y+3. Don't let the double negative fool you!

Recall Trigger

Think: 'manhole cover centered at (h,k) — always SUBTRACT the center coordinates'

Tags

• classification
• eccentricity
• conic sections
• key concept

Topic

Conic Sections

Concept

Eccentricity Classification of Conics: e=0 circle, e<1 ellipse, e=1 parabola, e>1 hyperbola

Anchor Id

A8

Difficulty

medium

Memory Aid

Use the phrase: 'ZERO Circle, LESS Ellipse, ONE Parabola, MORE Hyperbola' — arrange them on a number line from 0 to ∞. Think of eccentricity as 'how stretched' the curve is. A circle is perfectly round (e=0, perfectly unbiased). An ellipse is slightly stretched (e<1). A parabola is the tipping point (e=1). A hyperbola is completely split open (e>1). As eccentricity increases, the curve goes from 'closed and round' to 'wide open'.

Anchor Type

mnemonic

Why It Works

The number line visualization provides spatial memory. The progression from 'closed to open' maps directly onto the physical appearance of each conic.

Example Usage

Board question: 'Find eccentricity of x²/25 + y²/9 = 1.' Since a²=25, b²=9, c²=25−9=16, c=4, e=4/5=0.8. Since e<1, confirmed ELLIPSE.

Recall Trigger

Think: 'eccentricity number line: 0 = circle → 1 = parabola → beyond 1 = hyperbola'

Tags

• formula
• ellipse
• foci
• c squared

Topic

Conic Sections — Ellipse

Concept

Ellipse: c² = a² − b² (subtract for ellipse)

Anchor Id

A9

Difficulty

medium

Memory Aid

ELLIPSE = SUBTRACT. The letter 'E' in Ellipse stands for 'Every time we take Away' (subtract). c² = a² MINUS b². Alternatively: an ellipse is a CLOSED curve — it holds things IN. So the foci are INSIDE (c < a), meaning c must be LESS than a, so you SUBTRACT b² to make c² smaller. Hyperbola is the opposite: open and flying apart, so you ADD b² to make c² bigger.

Anchor Type

mnemonic

Why It Works

The verbal association 'Ellipse → subtract, Hyperbola → add' directly addresses the most common and costly board exam mistake. The 'closed vs open' spatial reasoning reinforces the rule.

Example Usage

Ellipse x²/25+y²/9=1: a²=25, b²=9, c²=25−9=16, c=4. Foci at (±4,0). Check: c=4 < a=5 ✓ (foci inside the ellipse).

Recall Trigger

Think: 'Ellipse = closed = subtract: c²=a²−b²'

Tags

• formula
• hyperbola
• foci
• c squared

Topic

Conic Sections — Hyperbola

Concept

Hyperbola: c² = a² + b² (add for hyperbola)

Anchor Id

A10

Difficulty

medium

Memory Aid

HYPERBOLA = ADD. Hyperbola has a 'PLUS' personality — it's dramatic, it flies apart in TWO directions, and c² = a² PLUS b². The foci are OUTSIDE (c > a), meaning c is BIGGER than a, so you ADD b² to make c² larger. Memory hook: 'Hyperbola is HYPER — it's EXTRA, it adds MORE (plus b²).' The word 'hyper' means excessive — so it adds, not subtracts.

Anchor Type

mnemonic

Why It Works

The 'hyper = excessive = adds more' wordplay is a strong semantic hook. Pairing it with the ellipse anchor (A9) as opposites strengthens both memories through contrast.

Example Usage

Hyperbola x²/16−y²/9=1: a²=16, b²=9, c²=16+9=25, c=5. Foci at (±5,0). Check: c=5 > a=4 ✓ (foci outside the vertices).

Recall Trigger

Think: 'Hyper = excessive = ADDS: c²=a²+b²'

Tags

• formula
• parabola
• focal distance
• 4a coefficient

Topic

Conic Sections — Parabola

Concept

Parabola: (y−k)² = 4a(x−h) — the 4a coefficient and focal distance

Anchor Id

A11

Difficulty

medium

Memory Aid

A satellite dish engineer (parabolic reflector) tells his apprentice: 'The coefficient of the linear term is always FOUR times the focal distance — never forget the FOUR!' He writes it on the board: 4a. If the board exam gives you (y²=8x), then 4a=8, so a=2. The focus is 2 units from the vertex. The trick: ALWAYS divide the coefficient by 4 to get the focal distance 'a'. The dish was designed for receiving signals exactly at the focus point — 'a' meters from the vertex.

Anchor Type

micro_story

Why It Works

The satellite dish is a real engineering application of parabolas. Emphasizing the division by 4 procedure directly prevents the common mistake of taking the coefficient itself as the focal distance.

Example Usage

Parabola y²=12x: 4a=12, a=3. Vertex at origin, focus at (3,0), directrix x=−3. Opens to the RIGHT (since x term is positive).

Recall Trigger

Think: 'satellite dish — coefficient ÷ 4 = focal distance a'

Tags

• classification
• identification
• general equation
• board exam

Topic

Conic Sections

Concept

Identifying Conic Sections from General Equation Ax²+Bxy+Cy²+Dx+Ey+F=0

Anchor Id

A12

Difficulty

hard

Memory Aid

Use 'CHESS PIECES CHECK': Circle → A=C (equal squared coefficients, no xy term). Ellipse → A≠C but same SIGN (both positive or both negative). Hyperbola → opposite SIGNS (one positive, one negative). Parabola → one squared term MISSING (A=0 or C=0). Acronym: CEHP — 'Circle Equal, Hyperbola Hostile (opposite signs), Parabola absent'. Or visually: circle is symmetric (A=C), ellipse is close to circle (A≈C, same sign), hyperbola is broken apart (opposite signs), parabola is incomplete (missing one square).

Anchor Type

acronym

Why It Works

Categorizing by coefficient relationships gives a systematic discrimination procedure. The 'CHESS PIECES' framing makes it feel like a game rule, reducing anxiety.

Example Usage

4x²+9y²−36=0: A=4, C=9, same sign, A≠C → ELLIPSE. 4x²−9y²−36=0: A=4, C=−9, opposite sign → HYPERBOLA. x²−4x+y=0: only one squared term → PARABOLA.

Recall Trigger

Think: 'CEHP — Equal, Same Sign, Opposite Sign, Missing'

Tags

• formula
• line equation
• general form
• conversion

Topic

Points and Lines

Concept

General Form of Line: Ax + By + C = 0

Anchor Id

A13

Difficulty

easy

Memory Aid

Think of the general form as a 'BALANCE SHEET' in accounting (familiar to Filipino engineers who deal with project budgets). Everything is on ONE SIDE — all terms equal ZERO. A is the x-account, B is the y-account, C is the constant overhead — and the total balance is always ZERO. This is why it's called 'general form' — it handles all lines including vertical lines (which slope-intercept form cannot handle: vertical line x=3 becomes 1x+0y−3=0).

Anchor Type

analogy

Why It Works

The balance sheet analogy is practical and familiar. Highlighting that general form handles vertical lines (where slope is undefined) addresses a key conceptual gap.

Example Usage

Convert y=4x/3+2/3 to general form: multiply by 3 → 3y=4x+2 → 4x−3y+2=0. Check: A=4, B=−3, C=2.

Recall Trigger

Think: 'balance sheet — everything on one side, equals zero'

Tags

• procedure
• perpendicular bisector
• midpoint
• slope

Topic

Points and Lines

Concept

Perpendicular Bisector of a Segment

Anchor Id

A14

Difficulty

medium

Memory Aid

Place yourself at the MIDPOINT of the segment (you are the bisector). You stand perpendicular to the segment — your orientation is 90° to it. TWO STEPS: (1) Find the midpoint — that's where YOU stand. (2) Find the perpendicular slope — FLIP AND NEGATE the original slope. Then write the line equation through the midpoint with the perpendicular slope. Loci: midpoint = your location; perpendicular slope = your direction.

Anchor Type

method_of_loci

Why It Works

Method of loci (memory palace) places the student physically at the key geometric location, making the two-step procedure feel natural and spatial.

Example Usage

Perpendicular bisector of (2,1) to (6,5): Midpoint=(4,3). Original slope m=(5−1)/(6−2)=1. Perp slope=−1. Line: y−3=−1(x−4) → x+y−7=0.

Recall Trigger

Think: 'I AM the perpendicular bisector — stand at midpoint, face perpendicularly'

Tags

• ellipse
• major axis
• orientation
• visual

Topic

Conic Sections — Ellipse

Concept

Ellipse Standard Form: x²/a² + y²/b² = 1, with a > b, major axis along x-axis

Anchor Id

A15

Difficulty

medium

Memory Aid

Visualize a FOOTBALL (American football / rugby ball) placed lying flat on the ground — it stretches HORIZONTALLY. The BIGGER denominator (a²) is under x², meaning the ellipse stretches in the x-DIRECTION. If a²=25 is under x², the semi-major axis is 5 in the x-direction. If a²=25 were under y², the football stands UPRIGHT — stretching vertically. RULE: The BIGGER denominator tells you which direction the major axis goes.

Anchor Type

visual_association

Why It Works

The football orientation is an immediate visual cue. The rule 'bigger denominator = direction of major axis' is a simple discriminating rule for a commonly confused concept.

Example Usage

x²/25+y²/9=1: a²=25 under x → horizontal major axis, a=5, b=3. Vertices at (±5,0). x²/9+y²/25=1: a²=25 under y → vertical major axis. Vertices at (0,±5).

Recall Trigger

Think: 'football — bigger denominator = direction of stretch'

Tags

• circle
• completing the square
• general form
• radius

Topic

Conic Sections — Circle

Concept

Radius of Circle from General Form x²+y²+Dx+Ey+F=0

Anchor Id

A16

Difficulty

hard

Memory Aid

Use 'COMPLETE THE SQUARE — D and E get halved and squared, then subtract F.' The formula: r = √((D/2)²+(E/2)²−F). Memory hook: 'D and E are HALF price at the sale (divide by 2), then SQUARE the discounted price, subtract the fixed cost F, and take the root to get the radius.' Always complete the square by grouping x-terms and y-terms before identifying center and radius.

Anchor Type

mnemonic

Why It Works

The 'half-price sale' analogy makes the halving operation memorable. Connecting it to completing the square reinforces the underlying algebraic procedure.

Example Usage

x²+y²−4x+6y−3=0: D=−4, E=6, F=−3. Center=(2,−3). r=√((2)²+(−3)²−(−3))=√(4+9+3)=√16=4.

Recall Trigger

Think: 'half-price sale on D and E, subtract F, take the root'

Tags

• formula
• angle between lines
• slopes
• trigonometry link

Topic

Points and Lines

Concept

Angle between Two Lines: tan θ = |(m₁−m₂)/(1+m₁m₂)|

Anchor Id

A17

Difficulty

hard

Memory Aid

Chunk it as: 'DIFFERENCE over SUM-OF-PRODUCTS, take absolute value, then arctan.' Numerator = m₁ MINUS m₂ (difference of slopes). Denominator = 1 PLUS m₁ times m₂ (one plus product of slopes). This is the tangent subtraction formula from trigonometry: tan(α−β) = (tan α − tan β)/(1 + tan α tan β). The slopes ARE the tangents of the inclination angles! Absolute value ensures acute angle.

Anchor Type

chunking

Why It Works

Recognizing the angle-between-lines formula as the trig tangent subtraction identity provides a deep conceptual anchor. Students who know trig identities will never forget this formula.

Example Usage

Angle between lines with slopes 2 and 1/3: tan θ = |(2−1/3)/(1+2×1/3)| = |(5/3)/(5/3)| = 1 → θ = 45°.

Recall Trigger

Think: 'tangent subtraction formula from trig — slopes are tangents of inclination angles'

Tags

• parabola
• directrix
• focus
• definition

Topic

Conic Sections — Parabola

Concept

Parabola Directrix: located at distance 'a' from vertex, opposite to the focus

Anchor Id

A18

Difficulty

medium

Memory Aid

Imagine a REFEREE (directrix) and a PLAYER (focus) on opposite sides of the GOAL LINE (vertex). Both stand exactly 'a' meters from the goal line (vertex), but on OPPOSITE SIDES. Every point on the parabola is equidistant from the focus (player) and the directrix (referee line). This is the very DEFINITION of a parabola. Vertex is the midpoint between focus and directrix.

Anchor Type

analogy

Why It Works

Sports analogies resonate with Filipino students. The physical symmetry of the referee-player positioning perfectly captures the equidistant property of the parabola.

Example Usage

Parabola y²=8x: a=2. Vertex (0,0). Focus at (2,0). Directrix: x=−2. Any point on curve: distance to focus = distance to directrix.

Recall Trigger

Think: 'referee and player, equal distance from goal line — focus and directrix, equal distance from vertex'

Tags

• line equation
• sign convention
• side determination
• applications

Topic

Points and Lines

Concept

Sign Convention: Line Ax+By+C=0 — which side of the line a point is on

Anchor Id

A19

Difficulty

hard

Memory Aid

Substitute the point into Ax+By+C. If the result is POSITIVE, the point is on the POSITIVE side (same side as the direction the normal vector points). If NEGATIVE, opposite side. In engineering surveying (traverse calculations), this tells you which side of a boundary line a structure falls on — critical for setback compliance in the NSCP. The absolute value gives DISTANCE; the sign gives SIDE. Always check both!

Anchor Type

visual_association

Why It Works

Connecting the sign interpretation to real-world NSCP setback compliance makes it professionally relevant. The dual interpretation (absolute = distance, sign = side) is a concise summary of the full power of the formula.

Example Usage

Point (0,0) vs line 3x+4y−10=0: 3(0)+4(0)−10=−10 (negative → origin is on the negative side). Point (5,5): 15+20−10=25 (positive → same side as normal).

Recall Trigger

Think: 'which side of the boundary? — positive result = same side as normal vector'

Tags

• hyperbola
• asymptotes
• slope
• b/a

Topic

Conic Sections — Hyperbola

Concept

Asymptotes of Hyperbola: y = ±(b/a)x for hyperbola centered at origin

Anchor Id

A20

Difficulty

hard

Memory Aid

For x²/a²−y²/b²=1: ASYMPTOTES are y=±(b/a)x. Notice: it's b/a, NOT a/b! Remember by the order in the equation — the x-term has 'a' in the denominator, the y-term has 'b'. The asymptote slope uses b OVER a (reverse order: y's coefficient over x's coefficient). Memory hook: 'HYPERBOLA ASYMPTOTE — go against the grain: b is on TOP even though a is the bigger one for the horizontal hyperbola.' For a vertical hyperbola (y²/a²−x²/b²=1), asymptotes are y=±(a/b)x.

Anchor Type

mnemonic

Why It Works

Explicitly flagging the b/a vs a/b confusion and providing a rule prevents a very common error. The 'reverse order' hook makes the exception memorable.

Example Usage

x²/16−y²/9=1: a²=16, b²=9, a=4, b=3. Asymptotes: y=±(3/4)x. These are the diagonals of the rectangle formed by the vertices and co-vertices.

Recall Trigger

Think: 'hyperbola asymptote slope = b over a (not a over b for horizontal hyperbola)'

Formula

d = √[(x₂−x₁)² + (y₂−y₁)²]

Mnemonic

DELTA-SQUARE-ADD-ROOT: Take Δx and Δy, SQUARE both, ADD them, take the SQUARE ROOT. Same as Pythagorean theorem — it IS the Pythagorean theorem.

When To Use

Whenever you need the straight-line distance between two known coordinate points. Appears in problems involving chord lengths, radii, segment lengths, and coordinate geometry setups.

What Each Part Means

Δx = x₂−x₁ is horizontal separation; Δy = y₂−y₁ is vertical separation; squaring makes both positive; adding combines both directions; root gives the straight-line hypotenuse distance.

Formula

Midpoint M = ((x₁+x₂)/2, (y₁+y₂)/2)

Mnemonic

AVERAGE BOTH: Take the arithmetic average of the x-coordinates and the arithmetic average of the y-coordinates. Just add and halve — no squaring, no roots.

When To Use

Finding the center of a segment, the center of a circle given diameter endpoints, or the midpoint for perpendicular bisector problems.

What Each Part Means

(x₁+x₂)/2 = average x-position; (y₁+y₂)/2 = average y-position. The midpoint is literally the 'average location' of the two endpoints.

Formula

m = (y₂−y₁)/(x₂−x₁)

Mnemonic

RISE over RUN — Y on TOP (numerator), X on BOTTOM (denominator). Y comes before X alphabetically, just as numerator comes before denominator.

When To Use

Any problem requiring slope of a line between two points, parallel/perpendicular conditions, or inclination angle (θ = arctan|m|).

What Each Part Means

Numerator = vertical change (rise); denominator = horizontal change (run). Positive m = goes uphill left to right. Negative m = goes downhill left to right. m=0 = horizontal. Undefined m = vertical (Δx=0).

Formula

d = |Ax₀+By₀+C| / √(A²+B²)

Mnemonic

PLUG-ABSOLUTE-DIVIDE: Plug (x₀,y₀) into line equation → take absolute value → divide by √(A²+B²). The denominator normalizes the line equation's scale.

When To Use

Finding the perpendicular distance from any point to any line. Also used in finding distances between parallel lines (apply to any point on one line).

What Each Part Means

Ax₀+By₀+C = how far the point is from satisfying the line equation (the 'residual'). |...| = distance is always positive. √(A²+B²) = magnitude of the normal vector to the line (the normalizer).

Formula

(x−h)²+(y−k)²=r²

Mnemonic

SUBTRACT THE CENTER, SQUARE BOTH, SET EQUAL TO r². The center coordinates SUBTRACT inside the parentheses. Expanding gives x²+y²−2hx−2ky+(h²+k²−r²)=0.

When To Use

Writing circle equations given center and radius. Identifying center and radius from standard form. Converting to/from general form.

What Each Part Means

(x−h) = horizontal distance from center; (y−k) = vertical distance from center; r² = radius squared. The equation says: the sum of squares of distances from center in x and y equals r².

Formula

Ellipse: x²/a²+y²/b²=1 with c²=a²−b²

Mnemonic

ELLIPSE SUBTRACTS: c²=a²MINUS b². The foci are INSIDE (c<a). Bigger denominator = direction of major axis. Eccentricity e=c/a is between 0 and 1.

When To Use

Any ellipse problem: finding foci, vertices, eccentricity, directrices, or latus rectum (LR=2b²/a).

What Each Part Means

a = semi-major axis (larger); b = semi-minor axis (smaller); c = focal distance (center to focus); e = c/a = shape descriptor. a²=b²+c² rearranged: Pythagorean relationship between semi-axes and focal distance.

Formula

Hyperbola: x²/a²−y²/b²=1 with c²=a²+b²

Mnemonic

HYPERBOLA ADDS: c²=a²PLUS b². The foci are OUTSIDE (c>a). Asymptotes y=±(b/a)x. Eccentricity e=c/a > 1.

When To Use

Hyperbola problems: foci, vertices, asymptotes, eccentricity, directrices, latus rectum (LR=2b²/a).

What Each Part Means

a = distance from center to vertex; b = determines spread of asymptotes; c = distance from center to focus; always c>a since b²>0 makes c²>a².

Formula

Parabola: (y−k)²=4a(x−h) or (x−h)²=4a(y−k)

Mnemonic

THE COEFFICIENT IS ALWAYS 4a — divide by 4 to get focal distance a. SQUARE TERM determines opening direction: y² → opens left/right; x² → opens up/down.

When To Use

Parabola problems: finding vertex, focus, directrix, latus rectum (LR=4a), axis of symmetry.

What Each Part Means

h,k = vertex; a = focal distance (vertex to focus = vertex to directrix); 4a = the full coefficient on the linear term. Focus at (h+a,k) for horizontal parabola opening right.

Formula

Angle of inclination: θ = arctan(m), 0° ≤ θ < 180°

Mnemonic

INCLINATION = ARCTANGENT OF SLOPE. For positive m → acute θ (0° to 90°). For negative m → obtuse θ (90° to 180°). Horizontal line: θ=0°. Vertical line: θ=90° (undefined slope).

When To Use

When a board problem asks for the angle a line makes with the x-axis, or when converting between angle and slope representations.

What Each Part Means

θ = angle line makes with positive x-axis, measured counterclockwise. m = tan θ. This is the 'tilt angle' of the line.

Chain Title

Four Conic Sections — Properties in Order

Recall Test

Without looking: What is the eccentricity range for each conic? What is c² equal to for ellipse vs hyperbola? What does the '4a' in a parabola represent?

Memory Chain

Story: A CIRCLE (perfectly round tire) rolls along a road and gradually FLATTENS into an ELLIPSE (slightly deformed tire). It keeps deforming until it becomes a PARABOLA (satellite dish mounted on a tower). Then it snaps into two pieces — a HYPERBOLA (two separate curves flying apart). Eccentricity climbs: 0 → less than 1 → exactly 1 → greater than 1.

Items To Remember

• Circle: e=0, c=0, equation (x−h)²+(y−k)²=r²
• Ellipse: e<1, c²=a²−b², sum of distances to foci is constant
• Parabola: e=1, equidistant from focus and directrix, coefficient = 4a
• Hyperbola: e>1, c²=a²+b², difference of distances to foci is constant

Chain Title

Steps to Find Distance from Point to Line

Recall Test

Apply to: point (2,3), line 4x+3y−12=0. What is the distance? Answer: |4(2)+3(3)−12|/√(16+9) = |8+9−12|/5 = 5/5 = 1.

Memory Chain

PHRASE: 'GENERAL PLUG ABSOLUTE NORMALIZE' — Get the line into General form, Plug in the point, take ABSOLute value, then NORMALIZE by dividing by √(A²+B²). Four words = four steps (after getting general form). Write them on your scratch paper at the start of the exam.

Items To Remember

• Step 1: Write line in general form Ax+By+C=0
• Step 2: Substitute point coordinates (x₀,y₀) into Ax₀+By₀+C
• Step 3: Take absolute value of the result
• Step 4: Divide by √(A²+B²)
• Step 5: Result is the perpendicular distance

Chain Title

Steps to Convert Circle from General to Standard Form

Recall Test

Convert x²+y²−6x+4y−3=0 to standard form. Answer: (x−3)²+(y+2)²=16. Center (3,−2), radius 4.

Memory Chain

Remember 'GROUP-HALF-SQUARE-TWICE': GROUP x and y terms. Take HALF of each linear coefficient. SQUARE those halves and add to both sides. Do this TWICE (once for x, once for y). The result is standard form. 'GROUP, HALF, SQUARE, TWICE' = complete the square procedure.

Items To Remember

• Step 1: Group x-terms and y-terms
• Step 2: Complete the square for x: add (D/2)² to both sides
• Step 3: Complete the square for y: add (E/2)² to both sides
• Step 4: Write as (x−h)²+(y−k)²=r²
• Step 5: Read off center (h,k) and radius r

Chain Title

Parallel and Perpendicular Line Conditions

Recall Test

Line L1: 2x−3y+1=0. Find the slope. Write the equation of a line through (1,2) that is: (a) parallel to L1, (b) perpendicular to L1. Answers: m_L1=2/3; (a) 2x−3y+4=0; (b) 3x+2y−7=0.

Memory Chain

Story: Two PARALLEL roads in BGC have the SAME slope (same grade) but different elevations (different y-intercepts). When you turn 90° onto a PERPENDICULAR street, the new slope is the FLIPPED AND NEGATED version of the original. You can always verify: multiply the two slopes — if you get −1, they are perpendicular.

Items To Remember

• Parallel: equal slopes, m₁=m₂
• Perpendicular: negative reciprocal slopes, m₁×m₂=−1
• Parallel lines have different y-intercepts (otherwise they are the same line)
• Perpendicular through a point: use FLIP AND NEGATE of original slope

Chain Title

Eccentricity Values Across All Conics

Recall Test

Quick quiz: e=0.6? e=1? e=2.5? e=0? Answers: ellipse, parabola, hyperbola, circle.

Memory Chain

Number line memory: '0, LESS-THAN-1, EXACTLY-1, GREATER-THAN-1.' Chant: 'ZERO circle, between ZERO and ONE is ellipse, ONE is parabola, MORE than ONE is hyperbola.' Eccentricity measures how 'off-center' the shape is — a circle has no offset (e=0), a hyperbola is completely off (e>1).

Items To Remember

• Circle: e = 0
• Ellipse: 0 < e < 1
• Parabola: e = 1
• Hyperbola: e > 1