· 3 min read

Coordinate Geometry Study Guide: 54 Flashcards from Slope to Conic Sections

Master coordinate geometry with 54 flashcards: lines, distance, conics, planes and polar coordinates, plus the 5 cards to fix and the worked examples to add.

Glowing Cartesian grid with a line, circle and parabola, compass and ruler among floating flashcards

Studying. But it's also a video game. 🎮

Flashcards, quizzes and games that make you actually remember.

Start playing — it's free

Posted by

Coordinate geometry turns shapes into equations: a line becomes y = mx + b, a circle becomes (x − h)² + (y − k)² = r², and "where do they meet?" becomes a system to solve. Flip It deck TJ9559 "Coordinate Geometry" packs 54 cards built from the classic analytic-geometry syllabus, from Descartes to tangent planes.

Study this deck: Coordinate Geometry

1 / 10
Question

Analytic geometry

Knew itLearning

Tap the card for the answer · swipe right if you knew it, left if not

What the deck covers

You'll memorize: ordered pairs and triples, polar, cylindrical and spherical coordinates, slope, x- and y-intercepts, slope-intercept, point-slope and general form, parallel and perpendicular lines, distance and midpoint formulas in 2D and 3D, the circle, the four conic sections, the conic discriminant B² − 4AC, ellipse, parabola and hyperbola standard forms, rectangular hyperbolas, quadric surfaces, plane equations (point-normal and general), parametric lines, the dot product, six graph transformations, tangent and normal lines, locus, and the Descartes vs Fermat story.

FrontBack
Cartesian → polarr = √(x² + y²), θ = arctan(y/x)
Point-slope form of a liney − y₁ = m(x − x₁)
Two lines perpendicularm₁ × m₂ = −1
Distance formula in 2Dd = √[(x₂ − x₁)² + (y₂ − y₁)²]
Standard form of a circle(x − h)² + (y − k)² = r²
Discriminant of a conicB² − 4AC, used to classify the conic type
General form of a planeAx + By + Cz + D = 0, (A, B, C) = normal vector
Reflect a graph across the y-axisy = f(−x)
La GéométrieDescartes' 1637 essay founding analytic geometry

Study it in four blocks

  1. Cards 1-9, coordinate systems. Drill the conversions both ways.
  2. Cards 10-23, lines, distance, midpoint, circles. The core of any exam.
  3. Cards 24-39, conics, quadrics, planes, vectors. The bridge to multivariable calculus.
  4. Cards 40-54, transformations, tangents, history. Cards 46, 53 and 54 are history and foundations: low priority if you study for a test.

Five cards to repair

  • Card 7: θ = arctan(y/x) only works for x > 0. For (−1, −1) it gives 45° instead of 225°. Add "adjust for the quadrant (atan2)".
  • Card 27: the discriminant is given, the rule is not. Add it: B² − 4AC < 0 ellipse (circle if B = 0 and A = C), = 0 parabola, > 0 hyperbola.
  • Card 17: m₁ × m₂ = −1 fails for a horizontal line and a vertical one, which are perpendicular but have no product of slopes. Add the exception.
  • Card 29: y = ax² opens upward only if a > 0, and (x − h)² = 4p(y − k) only if p > 0.
  • Card 9: (ρ, θ, φ) with θ in the xy-plane is the math convention. Physics textbooks swap θ and φ, so write your course's convention in the hint.

Tip: write the missing exercise cards

Not one card asks you to compute. This is a rule index, so add 5-6 worked cards: the distance from (1, 2) to (4, 6) (answer 5), the midpoint of (−2, 3) and (4, 7) (answer (1, 5)), the line through (1, 3) with slope 2, the center and radius of x² + y² − 6x + 4y − 3 = 0 (complete the square: (3, −2), r = 4). Three high-yield formulas are missing too: distance from a point to a line |Ax₀ + By₀ + C| / √(A² + B²), the shoelace area of a triangle, and the ellipse foci c² = a² − b².

For the shapes behind the formulas, see triangles and polygons and circle geometry. Find TJ9559 in the Flip It library.

Ready to learn faster? 🚀

Join FLIP IT for free and turn studying into a game.

Create my free account