GELE Mathematics — Analytic GeometryCheat Sheet
One-page cheat sheet for GELE Mathematics — Analytic Geometry. Every formula, definition, and key fact you need for this chapter, condensed to a single printable page. Designed for the final review session before the GELE 2026.
Exam context
The Geodetic Engineer Licensure Examination is conducted by Professional Regulation Commission (PRC) — Board of Geodetic Engineering and is scheduled for September 2026. The Mathematics subtest is marked as "Core" in the official pattern, and Analytic Geometry appears in position 4th of 10 in the GELE Mathematics review rotation. Passing mark: 70% weighted average, no sub-test below 50%. Recent GELE 2026 papers have drawn roughly a meaningful share of questions from this subject.
Analytic Geometry - Cheat Sheet
Your last-minute reference for points, lines, distances, and conic sections. Master coordinate geometry, line equations, and conic identification in 30 minutes.
Sections
Formulas
Formula
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Meaning
d = distance between two points; (x₁,y₁) and (x₂,y₂) are point coordinates
Watch Out
Square BOTH differences before adding; forgetting the square root is a common trap
When To Use
Always the first step when comparing two points; essential for surveying and layout calculations
Formula
Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Meaning
Coordinates of the point exactly halfway between two given points
Watch Out
This is not the average of distances — it's the average of coordinates separately
When To Use
Finding perpendicular bisectors, center of a segment, or symmetric points
Section Title
Points and Distance
Important Facts
- Origin is (0, 0)
- Distance is always positive; it's a scalar
- Points on the x-axis have y = 0; on y-axis have x = 0
- Distance formula is derived from Pythagorean theorem
Key Definitions
Term
Coordinate system
Example
Point (3, 5) is 3 units right and 5 units up from origin (0, 0)
Definition
A system using ordered pairs (x, y) to locate points on a plane, with x-axis horizontal and y-axis vertical.
Term
Quadrants
Example
Point (−2, 3) is in Quadrant II
Definition
Four regions: I (+,+), II (−,+), III (−,−), IV (+,−)
Diagrams To Know
- Coordinate plane with four quadrants labeled
- Two points and distance calculation illustrated
- Midpoint dividing a segment into two equal parts
Formulas
Formula
m = (y₂ - y₁)/(x₂ - x₁) = Δy/Δx
Meaning
m = slope (steepness); Δy = vertical change; Δx = horizontal change
Watch Out
Undefined when x₂ = x₁ (vertical line); zero when y₂ = y₁ (horizontal line). Order matters: (y₂ − y₁)/(x₂ − x₁), not reversed
When To Use
Every line problem; the foundation of line equations
Formula
Parallel lines: m₁ = m₂
Meaning
Two lines are parallel if and only if their slopes are equal
Watch Out
Vertical lines (undefined slope) are parallel to each other, but slope comparison doesn't apply directly
When To Use
Identifying or writing equations of parallel lines
Formula
Perpendicular lines: m₁ · m₂ = −1
Meaning
Two lines are perpendicular if product of slopes = −1 (slopes are negative reciprocals)
Watch Out
The slope is the NEGATIVE reciprocal, not just the reciprocal. If m₁ = 2/3, then m₂ = −3/2
When To Use
Right angles, perpendicular bisectors, orthogonal structures
Formula
y − y₁ = m(x − x₁)
Meaning
Point-slope form: m = slope; (x₁, y₁) = known point on line
Watch Out
Must substitute x₁ and y₁ (coordinates), not x and y (variables). Often rearranged to standard form
When To Use
When you have a slope and one point; easiest form to start with
Formula
y = mx + b
Meaning
Slope-intercept form: m = slope; b = y-intercept (where line crosses y-axis)
Watch Out
b is NOT the same as any coordinate; it's where x = 0. Be careful: if line doesn't cross y-axis (vertical), this form doesn't apply
When To Use
Graphing or when y-intercept is known; most familiar form
Formula
Ax + By + C = 0
Meaning
General/standard form: A, B, C are constants; typically A, B are integers with no common factor
Watch Out
Different sources use different sign conventions (±C); always rewrite to standard form for point-to-line distance. Ensure GCD(A,B,C) = 1 for 'standard' form
When To Use
Distance calculations, intersections, most formal/exam presentations
Common Values
Value
1
Symbol
m
Quantity
45° line slope
Value
1/√3 ≈ 0.577
Symbol
m
Quantity
30° line slope
Value
√3 ≈ 1.732
Symbol
m
Quantity
60° line slope
Section Title
Slope and Line Equations
Important Facts
- Horizontal lines have slope m = 0 and equation y = b
- Vertical lines have undefined slope and equation x = a
- Slope is independent of direction (same whether going left-to-right or right-to-left)
- Two distinct non-vertical lines with equal slope are parallel
- The line through origin (0,0) has no b term: y = mx
Key Definitions
Term
Slope
Example
Slope m = 2 means for every 1 unit right, line rises 2 units
Definition
Ratio of vertical rise to horizontal run; measure of steepness and direction of a line.
Term
Y-intercept
Example
Line y = 2x + 3 has y-intercept at (0, 3)
Definition
Point where line crosses the y-axis; always of form (0, b).
Term
X-intercept
Example
For y = 2x + 4, set y = 0 → x = −2, so x-intercept is (−2, 0)
Definition
Point where line crosses the x-axis; found by setting y = 0.
Diagrams To Know
- Slope triangle showing rise/run
- Parallel and perpendicular line pairs
- Line with marked intercepts
Reactions Or Equations
Note
Always multiply through to eliminate fractions before converting to general form
Equation
y − y₁ = m(x − x₁) ⟹ Ax + By + C = 0
Conditions
Convert point-slope to general form by expanding and rearranging
Formulas
Formula
d = |Ax₀ + By₀ + C| / √(A² + B²)
Meaning
d = perpendicular distance; (x₀, y₀) = point coordinates; Ax + By + C = 0 = line equation
Watch Out
Line MUST be in form Ax + By + C = 0. Use ABSOLUTE VALUE in numerator (distance is always positive). Denominator is √(A² + B²), NOT √(A + B)
When To Use
Finding shortest distance from a point to a line (perpendicular only); structural clearances, offset lines
Formula
Distance between parallel lines: d = |C₁ − C₂| / √(A² + B²)
Meaning
Both lines must have form Ax + By + C₁ = 0 and Ax + By + C₂ = 0 (same A and B)
Watch Out
Lines must have IDENTICAL A and B coefficients. If slopes are equal but A, B differ, multiply one equation to match first
When To Use
Parallel offset in surveying, construction tolerances, spacing calculations
Section Title
Distance from Point to Line
Important Facts
- The perpendicular from a point to a line intersects the line at exactly 90°
- This is the ONLY distance formula for point-to-line (not arc or diagonal distances)
- Both point and line must be in the same coordinate system (Cartesian, 2D)
- The perpendicular line through the point has slope = negative reciprocal of original line's slope
Key Definitions
Term
Perpendicular distance
Example
Distance from (3, 4) to line 3x + 4y − 10 = 0 is 3 units
Definition
The shortest possible distance from a point to a line, measured along a perpendicular.
Diagrams To Know
- Point and line with perpendicular dropped from point to line
- Parallel lines with distance marked between them
Formulas
Formula
(x − h)² + (y − k)² = r²
Meaning
(h, k) = center; r = radius; (x, y) = any point on circle
Watch Out
Signs: (x − h)² means center is at +h (not −h). Radius is r, NOT r². If given as x² + y² + Dx + Ey + F = 0, complete the square to find center and radius
When To Use
Identifying circles, writing circle equations, finding center and radius
Formula
Radius r = √(h² + k² − F) after completing square: (x − h)² + (y − k)² = h² + k² − F
Meaning
When circle equation is in general form x² + y² + Dx + Ey + F = 0
Watch Out
Complete the square SEPARATELY for x and y terms. The term being subtracted from (x − h)² + (y − k)² expansion gives r². Check r² > 0; if not, there's no real circle
When To Use
Converting from expanded form to standard form
Section Title
Circle
Important Facts
- Circle equation is a special case of ellipse (a = b = r)
- Circle passes through origin (0,0) ⟺ h² + k² = r² (center distance from origin = radius)
- A circle is completely defined by three non-collinear points
- Equation x² + y² = r² has center (0, 0) and radius r
Key Definitions
Term
Circle
Example
Circle centered at (2, 3) with radius 5 is (x − 2)² + (y − 3)² = 25
Definition
Locus of all points equidistant from a fixed point (center); distance is the radius.
Term
Tangent to circle
Example
The radius at tangent point is perpendicular to the tangent line
Definition
A line that touches the circle at exactly one point; perpendicular to radius at that point.
Diagrams To Know
- Circle with center (h, k), radius r marked
- Circle passing through three given points (circumcircle concept)
- Tangent line perpendicular to radius
Reactions Or Equations
Note
If D² + E² − 4F ≤ 0, the equation has no real solution (point or no locus)
Equation
x² + y² + Dx + Ey + F = 0 ⟹ (x + D/2)² + (y + E/2)² = (D² + E² − 4F)/4
Conditions
Complete the square for both variables; center is (−D/2, −E/2); radius² = (D² + E² − 4F)/4
Formulas
Formula
(y − k)² = 4a(x − h) [opens horizontally]
Meaning
(h, k) = vertex; a = focal distance from vertex; focus at (h+a, k); directrix x = h−a
Watch Out
The 4a is the COMPLETE coefficient (not just a). If a > 0, opens right; if a < 0, opens left. Directrix is a VERTICAL line
When To Use
Parabolas with horizontal axis; surveying sight lines, cable curves
Formula
(x − h)² = 4a(y − k) [opens vertically]
Meaning
(h, k) = vertex; a = focal distance; focus at (h, k+a); directrix y = k−a
Watch Out
If a > 0, opens upward; a < 0, opens downward. Directrix is a HORIZONTAL line. Ensure you identify which orientation is asked
When To Use
Parabolas with vertical axis; common in projectile motion, reflector dishes
Formula
Distance from any point on parabola to focus = distance to directrix
Meaning
Defining property: point P on parabola ⟹ |PF| = |PD| where F = focus, D = directrix
Watch Out
This is the DEFINITION; use it to verify or derive parabola equations, not as a formula for calculation alone
When To Use
Verifying points on parabola, constructing parabola geometrically
Section Title
Parabola
Important Facts
- Parabola equation: y² = 4ax has vertex (0,0), focus (a,0), directrix x = −a
- Parabola equation: x² = 4ay has vertex (0,0), focus (0,a), directrix y = −a
- Focal chord (chord through focus) has length ≥ 4a (minimum = latus rectum = 4a, perpendicular to axis)
- Parabola eccentricity e = 1 (always)
Key Definitions
Term
Vertex
Example
Parabola y² = 8x has vertex (0, 0), focus (2, 0), directrix x = −2
Definition
The turning point of the parabola; midpoint between focus and directrix.
Term
Focus
Example
For (y − k)² = 4a(x − h), focus is at (h + a, k)
Definition
Fixed point on the axis of symmetry; distance a from vertex.
Term
Directrix
Example
For (y − k)² = 4a(x − h), directrix is x = h − a
Definition
Fixed line perpendicular to axis of symmetry; distance a from vertex on opposite side of focus.
Term
Axis of symmetry
Example
For (x − h)² = 4a(y − k), axis is vertical line x = h
Definition
Line through vertex and focus; parabola is symmetric about this axis.
Diagrams To Know
- Parabola with vertex, focus, directrix, and axis of symmetry labeled
- Horizontal and vertical parabola orientations side-by-side
- Focal chord (latus rectum) drawn perpendicular to axis
Reactions Or Equations
Note
Coefficient 4a is critical: if given (y−k)² = 8(x−h), then 4a = 8 ⟹ a = 2
Equation
(y − k)² = 4a(x − h) ⟹ vertex (h,k), focus (h+a,k), directrix x=h−a
Conditions
Horizontal opening; a is the signed focal distance
Formulas
Formula
(x − h)²/a² + (y − k)²/b² = 1 [a > b, major axis horizontal]
Meaning
(h, k) = center; a = semi-major axis; b = semi-minor axis; c² = a² − b²; c = focal distance
Watch Out
ALWAYS a > b (major denominator is larger). Foci are at (h±c, k), NOT (h, k±c). c² = a² − b² (subtract, NOT add)
When To Use
Ellipses with horizontal major axis
Formula
(x − h)²/b² + (y − k)²/a² = 1 [a > b, major axis vertical]
Meaning
(h, k) = center; a = semi-major axis (vertical); b = semi-minor axis (horizontal); c² = a² − b²
Watch Out
Now the larger denominator is under (y−k)². Foci still satisfy c² = a² − b² and are at (h, k±c)
When To Use
Ellipses with vertical major axis
Formula
Eccentricity e = c/a where 0 < e < 1
Meaning
e measures how 'stretched' ellipse is; e = 0 is circle, e → 1 is very elongated
Watch Out
Always 0 < e < 1 for ellipse (never 0 or 1). e = c/a (NOT c/b). As b → a (circle), e → 0
When To Use
Comparing ellipses, orbital mechanics (elliptical orbits)
Section Title
Ellipse
Important Facts
- Sum of distances from any point on ellipse to both foci is always 2a (constant)
- Ellipse is symmetric about both major and minor axes
- If a = b, ellipse becomes a circle with c = 0 and e = 0
- For x²/a² + y²/b² = 1 with a > b: major axis is horizontal, foci on x-axis
- Vertices are endpoints of axes: (±a, 0) on major, (0, ±b) on minor (for standard horizontal ellipse)
Key Definitions
Term
Ellipse
Example
Ellipse x²/25 + y²/9 = 1 has semi-major axis a=5, semi-minor axis b=3, foci at (±4, 0)
Definition
Locus of points where sum of distances to two fixed points (foci) is constant = 2a.
Term
Major axis
Example
For x²/25 + y²/9 = 1, major axis is horizontal with length 2(5) = 10
Definition
Longest diameter through both foci; length = 2a.
Term
Minor axis
Example
For x²/25 + y²/9 = 1, minor axis is vertical with length 2(3) = 6
Definition
Shortest diameter perpendicular to major axis; length = 2b.
Term
Foci
Example
Ellipse x²/25 + y²/9 = 1: c² = 25 − 9 = 16 ⟹ c = 4, foci at (±4, 0)
Definition
Two fixed points on the major axis at distance c from center, where c² = a² − b².
Diagrams To Know
- Ellipse with center, vertices, foci, major and minor axes all labeled
- Two focal points with sum-of-distances property illustrated
- Horizontal and vertical ellipse orientations compared
Reactions Or Equations
Note
Always c < a because b² > 0; thus a² − b² < a²
Equation
c² = a² − b² (for ellipse)
Conditions
SUBTRACT (unlike hyperbola which adds); c is focal distance from center
Formulas
Formula
(x − h)²/a² − (y − k)²/b² = 1 [opens horizontally]
Meaning
(h, k) = center; a = semi-major (real) axis; b = semi-minor (imaginary) axis; c² = a² + b² (ADDITION)
Watch Out
c² = a² + b² (ADD, NOT subtract). Foci at (h±c, k). The − sign between terms means hyperbola opens left-right, NOT up-down
When To Use
Hyperbolas with horizontal transverse axis; asymptotes slope = ±b/a
Formula
(y − k)²/a² − (x − h)²/b² = 1 [opens vertically]
Meaning
(h, k) = center; a = semi-transverse (vertical) axis; c² = a² + b²; foci at (h, k±c)
Watch Out
Same c² = a² + b² rule. The first term is (y−k)²/a² (opens up-down). Asymptotes are y − k = ±(a/b)(x − h)
When To Use
Hyperbolas with vertical transverse axis; asymptotes slope = ±a/b
Formula
Eccentricity e = c/a where e > 1
Meaning
For hyperbola, e > 1 always; as e increases, hyperbola opens wider
Watch Out
e > 1 for hyperbola (DIFFERENT from ellipse where e < 1). Since c > a, we have e = c/a > 1
When To Use
Comparing hyperbolas, identifying conic type
Formula
Asymptotes for (x−h)²/a² − (y−k)²/b² = 1: y − k = ±(b/a)(x − h)
Meaning
Lines that the hyperbola branches approach but never touch
Watch Out
For horizontal hyperbola, slope = b/a. For vertical hyperbola, slope = a/b (REVERSED). The asymptotes pass through center (h, k)
When To Use
Sketching hyperbola; identifying direction and spread
Section Title
Hyperbola
Important Facts
- Hyperbola has TWO separate branches (unlike ellipse)
- Difference of distances from any point to both foci is always 2a (constant)
- Hyperbola is symmetric about both transverse and conjugate axes
- Vertices are the closest points to center: (±a, 0) for horizontal, (0, ±a) for vertical
- c² = a² + b² (ADDITION, opposite of ellipse)
- If a = b, hyperbola is rectangular/equilateral with e = √2
Key Definitions
Term
Hyperbola
Example
x²/9 − y²/16 = 1 has semi-real axis a=3, semi-imaginary axis b=4, foci at (±5, 0)
Definition
Locus of points where DIFFERENCE of distances to two fixed points (foci) is constant = 2a.
Term
Transverse axis
Example
For x²/a² − y²/b² = 1, transverse axis is horizontal
Definition
The axis connecting the two branches (through both vertices and foci); length = 2a.
Term
Conjugate axis
Example
For x²/a² − y²/b² = 1, conjugate axis is vertical
Definition
The axis perpendicular to transverse axis, through center; length = 2b (imaginary).
Term
Asymptote
Example
Hyperbola x²/9 − y²/4 = 1 has asymptotes y = ±(2/3)x
Definition
A line that the hyperbola curve approaches infinitely but never touches.
Diagrams To Know
- Hyperbola with two branches, center, vertices, foci, transverse/conjugate axes, asymptotes
- Horizontal and vertical hyperbola branches illustrated
- Asymptotes forming an X through the center
Reactions Or Equations
Note
Always c > a because b² > 0; thus a² + b² > a²
Equation
c² = a² + b² (for hyperbola)
Conditions
ADD (unlike ellipse which subtracts); c is focal distance from center
Formulas
Formula
e = 0: Circle
Meaning
Eccentricity zero indicates a perfect circle
Watch Out
Circle is a degenerate ellipse where a = b; standard form has equal denominators
When To Use
Identifying conic from eccentricity value
Formula
0 < e < 1: Ellipse
Meaning
Eccentricity between 0 and 1 indicates an ellipse
Watch Out
The closer e is to 1, the more elongated the ellipse. c² = a² − b² (subtract)
When To Use
Orbital paths, construction, geometric problems
Formula
e = 1: Parabola
Meaning
Eccentricity exactly 1; ratio of distance to focus vs directrix is 1
Watch Out
Only one focus, no second branch. Focus and directrix define the parabola uniquely
When To Use
Identifying parabola; trajectory problems
Formula
e > 1: Hyperbola
Meaning
Eccentricity greater than 1 indicates a hyperbola
Watch Out
Two branches opening away from center. c² = a² + b² (add). As e → ∞, branches become more open
When To Use
Identifying hyperbola; asymptotic behavior
Common Values
Value
0
Symbol
e
Quantity
Circle eccentricity
Value
0.2 to 0.9
Symbol
e
Quantity
Ellipse eccentricity (typical)
Value
1
Symbol
e
Quantity
Parabola eccentricity
Value
1.5 to 3
Symbol
e
Quantity
Hyperbola eccentricity (typical)
Section Title
Conic Section Identification and Eccentricity
Important Facts
- General conic form: Ax² + Bxy + Cy² + Dx + Ey + F = 0 (B² − 4AC determines type)
- If B² − 4AC < 0: ellipse (or circle if A = C, B = 0)
- If B² − 4AC = 0: parabola
- If B² − 4AC > 0: hyperbola
- All conics except parabola have two foci; parabola has one focus and directrix
- Center of conic (h, k) found by completing the square or using −D/2A, −E/2C
Key Definitions
Term
Conic section
Example
Cut horizontally = circle; at angle = ellipse; parallel to slant = parabola; through both cones = hyperbola
Definition
Curve formed by intersecting a plane with a double cone; includes circle, ellipse, parabola, hyperbola.
Term
Eccentricity
Example
Circle e=0, ellipse e=0.8, parabola e=1, hyperbola e=2
Definition
Parameter e = c/a measuring deviation from circle; determines conic type.
Term
Degenerate conic
Example
x² − y² = 0 factors as (x−y)(x+y) = 0, two intersecting lines
Definition
Special case: intersecting lines (hyperbola), single line (parabola), point (ellipse), or no real locus.
Diagrams To Know
- All four conic sections (circle, ellipse, parabola, hyperbola) with eccentricity labels
- Classification flowchart based on B² − 4AC discriminant
- Cone being cut at various angles to produce each conic
Must Remember
- Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²] — BOTH differences squared, then summed, then square-rooted. Forgetting the square root is fatal.
- Perpendicular slopes: m₁ · m₂ = −1 (negative reciprocal, NOT just reciprocal). If m₁ = 2/3, then m₂ = −3/2.
- Point-to-line distance: d = |Ax₀+By₀+C|/√(A²+B²) — Line MUST be in form Ax+By+C=0; use ABSOLUTE VALUE in numerator; denominator is √(A²+B²) not √(A+B).
- Parabola: (y−k)² = 4a(x−h) — The coefficient is 4a (as a UNIT), so if given (y−k)² = 8(x−h), then 4a = 8 ⟹ a = 2. Focus is at (h+a, k); directrix is x = h−a.
- Ellipse vs Hyperbola c-formula: Ellipse c² = a²−b² (SUBTRACT); Hyperbola c² = a²+b² (ADD). This is the #1 confusing point.
- Ellipse: a is ALWAYS the larger semi-axis (under the larger denominator); foci lie on the major axis. If x denominator > y denominator, major axis is horizontal.
- Hyperbola: Opens in direction of POSITIVE term. If (x−h)²/a² − (y−k)²/b² = 1, opens horizontally (left-right). If (y−k)²/a² − (x−h)²/b² = 1, opens vertically (up-down).
- Conic eccentricity: e=0 (circle), 0<e<1 (ellipse), e=1 (parabola), e>1 (hyperbola). Identifies conic type immediately.
- Line equations: Point-slope y−y₁ = m(x−x₁) is easiest to START with; slope-intercept y = mx+b is for graphing; general Ax+By+C=0 is standard form for distance calculations.
- Completing the square: x² + Dx ⟹ (x + D/2)² − D²/4. Do this separately for x and y to convert general form to standard form and find center/vertex.
Last Minute Tips
- READ THE EQUATION FORM CAREFULLY. An equation written as (x−2)² + (y+3)² = 9 has CENTER (2, −3) [note the sign flip on k], NOT (−2, +3). This sign-flip is a board-exam favorite.
- When given a general conic Ax² + Bxy + Cy² + Dx + Ey + F = 0, IMMEDIATELY compute B²−4AC to identify the type (< 0 ellipse/circle, = 0 parabola, > 0 hyperbola) before solving anything else.
- For parallel lines offset problems (common in surveying): Convert both lines to the form Ax + By + C₁ = 0 and Ax + By + C₂ = 0 with IDENTICAL A and B. If they differ, multiply one equation to match. Then distance = |C₁−C₂|/√(A²+B²).
- Always verify your conic type by checking if the equation makes geometric sense. For example, if you get r² < 0 for a circle, there's NO real circle — you've made an algebra error. Hyperbola with negative a² is impossible.
- In exam, DRAW a sketch (even rough) of the conic if interpreting real-world problems (cable stays, sight lines, structure offsets). A visual check catches sign errors and incorrect asymptote slopes instantly.
Comparison Tables
Rows
Values
- (x−h)²/a² + (y−k)²/b² = 1
- (x−h)²/a² − (y−k)²/b² = 1
Property
Standard form (horizontal)
Values
- c² = a² − b² (subtract)
- c² = a² + b² (add)
Property
Relationship between a, b, c
Values
- 0 < e < 1
- e > 1
Property
Eccentricity e
Values
- One closed curve
- Two separate branches
Property
Number of branches
Values
- Two (inside ellipse on major axis)
- Two (outside hyperbola on transverse axis)
Property
Number of foci
Values
- None
- Yes: y − k = ±(b/a)(x − h) [horizontal] or ±(a/b)(x − h) [vertical]
Property
Asymptotes
Values
- Sum of distances to foci = 2a (constant)
- Difference of distances to foci = 2a (constant)
Property
Sum or difference property
Values
- Becomes more circular (e → 0)
- Becomes equilateral (e → √2 ≈ 1.414)
Property
As b → a
Columns
- Feature
- Ellipse
- Hyperbola
Table Title
Ellipse vs Hyperbola — Key Differences
Rows
Values
- Right
- (0, 0)
- (a, 0)
- x = −a
- Horizontal (x-axis)
Property
y² = 4ax (a > 0)
Values
- Left
- (0, 0)
- (−a, 0)
- x = a
- Horizontal (x-axis)
Property
y² = −4ax (a > 0)
Values
- Up
- (0, 0)
- (0, a)
- y = −a
- Vertical (y-axis)
Property
x² = 4ay (a > 0)
Values
- Down
- (0, 0)
- (0, −a)
- y = a
- Vertical (y-axis)
Property
x² = −4ay (a > 0)
Columns
- Form
- Opens
- Vertex
- Focus
- Directrix
- Axis
Table Title
Parabola Types — Vertex (0,0) Reference
Rows
Values
- B² − 4AC < 0
- A = C, B = 0
- e = 0
Property
Circle
Values
- B² − 4AC < 0
- A ≠ C or B ≠ 0, same sign
- 0 < e < 1
Property
Ellipse
Values
- B² − 4AC = 0
- One squared term vanishes after completing square
- e = 1
Property
Parabola
Values
- B² − 4AC > 0
- A and C opposite signs
- e > 1
Property
Hyperbola
Columns
- Conic Type
- Discriminant B² − 4AC
- Coefficient Pattern
- Eccentricity e
Table Title
Conic Identification — General Equation Ax² + Bxy + Cy² + Dx + Ey + F = 0
Rows
Values
- 0
- Horizontal line y = b
Property
0°
Values
- 1/√3 ≈ 0.577
- Shallow upward line
Property
30°
Values
- 1
- y = x (diagonal)
Property
45°
Values
- √3 ≈ 1.732
- Steep upward line
Property
60°
Values
- Undefined
- Vertical line x = a
Property
90°
Columns
- Angle θ
- Slope m = tan(θ)
- Example
Table Title
Line Slopes — Common Angles
Ready to practise for the GELE 2026?
Super Tutor's AI review plan adapts to your weak areas and builds a weekly practice schedule around your target GELE exam date.