📊 Coordinate Geometry — Reference

Essential concepts and definitions for working with coordinate geometry in MATHCOUNTS.

Coordinate System

Basic Concepts

Origin: Point (0, 0) where x-axis and y-axis intersect x-axis: Horizontal line, y = 0 y-axis: Vertical line, x = 0 Quadrants: Four regions divided by axes

  • Quadrant I: x > 0, y > 0
  • Quadrant II: x < 0, y > 0
  • Quadrant III: x < 0, y < 0
  • Quadrant IV: x > 0, y < 0

Coordinates

Ordered pair: (x, y) where x is horizontal coordinate, y is vertical coordinate x-coordinate: Distance from y-axis (left/right) y-coordinate: Distance from x-axis (up/down)

Distance and Midpoint

Distance Formula

Formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²] Derivation: Pythagorean theorem Example: Distance between (1, 2) and (4, 6) = √[(4-1)² + (6-2)²] = √[9 + 16] = 5

Midpoint Formula

Formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2) Derivation: Average of coordinates Example: Midpoint of (1, 2) and (4, 6) = ((1+4)/2, (2+6)/2) = (2.5, 4)

Lines

Slope

Definition: Steepness of a line Formula: m = (y₂ - y₁)/(x₂ - x₁) Horizontal line: m = 0 Vertical line: m = undefined Positive slope: Line rises from left to right Negative slope: Line falls from left to right

Equation of a Line

Slope-intercept form: y = mx + b

  • m = slope, b = y-intercept Point-slope form: y - y₁ = m(x - x₁)
  • m = slope, (x₁, y₁) = point on line Standard form: Ax + By = C
  • A, B, C are integers, A > 0 Two-point form: (y - y₁)/(y₂ - y₁) = (x - x₁)/(x₂ - x₁)

Line Relationships

Parallel lines: Same slope (m₁ = m₂) Perpendicular lines: Slopes are negative reciprocals (m₁ × m₂ = -1) Intersecting lines: Different slopes Coincident lines: Same slope and y-intercept

Circles

Circle Equation

Standard form: (x - h)² + (y - k)² = r²

  • (h, k) = center, r = radius General form: x² + y² + Dx + Ey + F = 0
  • Center: (-D/2, -E/2)
  • Radius: r = √(D²/4 + E²/4 - F)

Circle Properties

Center: (h, k) Radius: r Diameter: 2r Circumference: 2πr Area: πr²

Circle Relationships

Point on circle: (x - h)² + (y - k)² = r² Point inside circle: (x - h)² + (y - k)² < r² Point outside circle: (x - h)² + (y - k)² > r²

Parabolas

Parabola Equations

Vertex form: y = a(x - h)² + k

  • (h, k) = vertex, a = direction and width Standard form: y = ax² + bx + c
  • Vertex: (-b/2a, c - b²/4a) Factored form: y = a(x - r₁)(x - r₂)
  • r₁, r₂ = x-intercepts

Parabola Properties

Vertex: (h, k) Axis of symmetry: x = h Focus: (h, k + 1/4a) Directrix: y = k - 1/4a Direction: Opens up if a > 0, down if a < 0

Ellipses

Ellipse Equation

Standard form: (x - h)²/a² + (y - k)²/b² = 1

  • (h, k) = center, a = semi-major axis, b = semi-minor axis Horizontal ellipse: a > b Vertical ellipse: b > a

Ellipse Properties

Center: (h, k) Vertices: (h ± a, k) and (h, k ± b) Foci: (h ± c, k) where c² = a² - b² Eccentricity: e = c/a

Hyperbolas

Hyperbola Equation

Standard form: (x - h)²/a² - (y - k)²/b² = 1

  • (h, k) = center, a = distance to vertices, b = distance to co-vertices Horizontal hyperbola: Opens left and right Vertical hyperbola: Opens up and down

Hyperbola Properties

Center: (h, k) Vertices: (h ± a, k) Foci: (h ± c, k) where c² = a² + b² Asymptotes: y - k = ±(b/a)(x - h)

Transformations

Translations

Horizontal: (x, y) → (x + a, y) Vertical: (x, y) → (x, y + b) Both: (x, y) → (x + a, y + b)

Reflections

Over x-axis: (x, y) → (x, -y) Over y-axis: (x, y) → (-x, y) Over y = x: (x, y) → (y, x) Over y = -x: (x, y) → (-y, -x)

Rotations

90° counterclockwise: (x, y) → (-y, x) 180°: (x, y) → (-x, -y) 270° counterclockwise: (x, y) → (y, -x) 90° clockwise: (x, y) → (y, -x)

Dilations

Center at origin: (x, y) → (kx, ky) Center at (a, b): (x, y) → (a + k(x - a), b + k(y - b))

Conic Sections

Circle

Equation: (x - h)² + (y - k)² = r² Properties: All points equidistant from center

Parabola

Equation: y = a(x - h)² + k Properties: All points equidistant from focus and directrix

Ellipse

Equation: (x - h)²/a² + (y - k)²/b² = 1 Properties: Sum of distances to foci is constant

Hyperbola

Equation: (x - h)²/a² - (y - k)²/b² = 1 Properties: Difference of distances to foci is constant

Common Applications

Real-world Problems

Navigation: Use coordinates for location Physics: Use coordinates for motion Engineering: Use coordinates for design Computer graphics: Use coordinates for images

Problem-solving Strategies

Plot points: Visualize the problem Use formulas: Apply appropriate formulas Check answers: Verify results make sense Use symmetry: Look for patterns

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