📍

Coordinate Geometry

54 cards · 🇺🇸 English

Points, slope, distance and lines in the coordinate plane.

geometrycoordinate-geometry

Study all 54 cards for free

Flashcards, quizzes, pixel-art games and a daily review that remembers for you.

Study this deck free

Try the first 10 cards

1 / 10
Question

Analytic geometry

Knew itLearning

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

Already have an account? Sign in

Cards

  1. 1Analytic geometry
    The study of geometry using a coordinate system, also known as coordinate or Cartesian geometry
  2. 2Cartesian coordinate system in 2D
    A coordinate system where each point has an x-coordinate (horizontal) and y-coordinate (vertical), written as (x, y)
  3. 3Ordered pair
    A pair of coordinates (x, y) representing a point in 2D Cartesian space
  4. 4Cartesian coordinate system in 3D
    A system where every point is represented by an ordered triple (x, y, z)
  5. 5Polar coordinates
    A 2D coordinate system where each point is represented by (r, θ): distance from origin and angle from positive x-axis
  6. 6from polar (r, θ) → Cartesian (x, y)
    x = r cos θ, y = r sin θ
  7. 7from Cartesian (x, y) → polar (r, θ)
    r = √(x² + y²), θ = arctan(y/x)
  8. 8Cylindrical coordinates
    3D coordinates (z, r, θ): height z, radius r from z-axis, and angle θ on xy-plane
  9. 9Spherical coordinates
    3D coordinates (ρ, θ, φ): distance ρ from origin, angle θ on xy-plane, angle φ from z-axis
  10. 10Slope-intercept form of a line
    y = mx + b, where m is the slope and b is the y-intercept
  11. 11Slope of a line
    The ratio of vertical change to horizontal change: m = (y₂ - y₁) / (x₂ - x₁)
  12. 12Y-intercept of a line
    The point where the line crosses the y-axis (where x = 0)
  13. 13X-intercept of a line
    The point where the line crosses the x-axis (where y = 0)
  14. 14General form of a linear equation
    Ax + By + C = 0
  15. 15Point-slope form of a line
    y - y₁ = m(x - x₁), where m is slope and (x₁, y₁) is a known point
  16. 16Two lines to be parallel
    They have equal slopes (m₁ = m₂) and never intersect
  17. 17Two lines to be perpendicular
    Their slopes are negative reciprocals: m₁ × m₂ = −1
  18. 18Distance formula in 2D
    d = √[(x₂ - x₁)² + (y₂ - y₁)²], derived from the Pythagorean theorem
  19. 19Distance formula in 3D
    d = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]
  20. 20Midpoint formula in 2D
    M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
  21. 21Midpoint formula in 3D
    M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
  22. 22Standard form equation of a circle
    (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius
  23. 23Equation of a circle centered at the origin
    x² + y² = r², where r is the radius
  24. 24Conic section
    A curve formed by the intersection of a plane and a double cone (circle, ellipse, parabola, hyperbola)
  25. 25Four types of conic sections
    Circle, ellipse, parabola, and hyperbola (plus degenerate cases)
  26. 26General form of a conic section equation
    Ax² + Bxy + Cy² + Dx + Ey + F = 0
  27. 27Discriminant of a conic section
    B² − 4AC, used to classify the conic type
  28. 28Standard form of an ellipse centered at origin
    x²/a² + y²/b² = 1, where a and b are semi-major and semi-minor axes
  29. 29Standard form of a parabola opening upward
    y = ax² or (x - h)² = 4p(y - k), where (h, k) is the vertex
  30. 30Standard form of a hyperbola centered at origin
    x²/a² − y²/b² = 1 (opens horizontally) or y²/a² − x²/b² = 1 (opens vertically)

And 24 more cards. Study the whole deck free

Articles about this deck

More Mathematics decks

See all
Study all 54 cards free