Coordinate Geometry Basics

Study Notes

Coordinate Geometry

Definition: A branch of mathematics that studies geometric figures using a coordinate system.
Coordinate System:
- Cartesian Coordinates: Uses a pair of numbers (x, y) to represent points on a plane.
  - Origin: The point (0, 0).
  - Axes:
    - x-axis (horizontal)
    - y-axis (vertical)
- Distance Formula:
  - Distance between two points ((x_1, y_1)) and ((x_2, y_2)):
    - (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})
Midpoint Formula:
- The midpoint M of a segment connecting points ((x_1, y_1)) and ((x_2, y_2)):
  - (M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))
Slope of a Line:
- The slope (m) between two points:
  - (m = \frac{y_2 - y_1}{x_2 - x_1})
- Identifies line steepness and direction:
  - Positive slope: Rising line
  - Negative slope: Falling line
  - Zero slope: Horizontal line
  - Undefined slope: Vertical line
Equation of a Line:
- Slope-intercept form:
  - (y = mx + b) where (m) is slope and (b) is y-intercept.
- Point-slope form:
  - (y - y_1 = m(x - x_1))
- Standard form:
  - (Ax + By = C)
Types of Lines:
- Parallel Lines: Same slope, never intersect.
- Perpendicular Lines: Slopes are negative reciprocals ((m_1 \cdot m_2 = -1)).
Conic Sections:
- Circle:
  - Standard equation: ((x - h)^2 + (y - k)^2 = r^2) where ((h, k)) is the center and (r) is the radius.
- Ellipse:
  - Standard equation: (\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1)
- Parabola:
  - Standard equation: (y = ax^2 + bx + c)
- Hyperbola:
  - Standard equation: (\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1)
Applications:
- Used in physics, engineering, computer graphics, and navigation.
Tools:
- Graphing calculators and software for visual representation.
- Coordinate grids for plotting and analyzing geometric properties.

Coordinate Geometry

A mathematical discipline that interprets geometric figures through a coordinate framework.

Coordinate System

Cartesian Coordinates: Employs (x, y) pairs to indicate positions on a plane.
Origin: Located at (0, 0), serving as the reference point.
Axes:
- x-axis: horizontal line
- y-axis: vertical line

Distance Formula

To compute the distance (d) between two coordinates ((x_1, y_1)) and ((x_2, y_2)):
- Formula: (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2})

Midpoint Formula

Calculates the midpoint (M) of a line segment connecting ((x_1, y_1)) and ((x_2, y_2)):
- Formula: (M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right))

Slope of a Line

The slope (m) indicates the angle of inclination between two coordinates:
- Formula: (m = \frac{y_2 - y_1}{x_2 - x_1})
Interpretations:
- Positive slope: rising line
- Negative slope: falling line
- Zero slope: horizontal line
- Undefined slope: vertical line

Equation of a Line

Slope-intercept form: (y = mx + b) with (m) as slope and (b) as y-intercept.
Point-slope form: (y - y_1 = m(x - x_1)).
Standard form: (Ax + By = C) where (A), (B), and (C) are constants.

Types of Lines

Parallel Lines: Lines sharing the same slope but never meet.
Perpendicular Lines: Lines that intersect at right angles; their slopes are negative reciprocals ((m_1 \cdot m_2 = -1)).

Conic Sections

Circle: Standard equation ((x - h)^2 + (y - k)^2 = r^2) identifies center ((h, k)) and radius (r).
Ellipse: Given by (\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1), describes an elongated circle.
Parabola: Standard form (y = ax^2 + bx + c) depicts a U-shaped curve.
Hyperbola: Standard equation (\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1) forms two separate curves.