Coordinate geometry

Coordinate Geometry

Basic Terminology

  • Coordinate geometry, also known as analytical geometry, is a branch of geometry where algebra is used to investigate geometric properties.
  • The Cartesian coordinate system is a 2D grid system which marks every point in a plane with a pair of numerical coordinates.
  • Coordinates are ordered pairs of numbers or values (x and y) that determine the position of points on a plane.
  • The x-coordinate (abscissa) is the horizontal value and the y-coordinate (ordinate) is the vertical value.
  • The point where the x and y axes intersect is called the origin and it has coordinates (0,0).
  • The horizontal axis is the x-axis and the vertical axis is the y-axis.

Key Concepts

  • A line can be defined by two points. The equation of a line is in the form y = mx + c where m is the slope and c is the y-intercept.
  • The slope (gradient) of a line is given by the formula m = (y2 - y1) / (x2 - x1) where (x1, y1) and (x2, y2) are two points on the line.
  • The y-intercept is the point where the line crosses the y-axis.
  • The midpoint between two points can be found using the formula ((x1 + x2)/2, (y1 + y2)/2).
  • The distance between two points can be found using the formula √((x2-x1)² + (y2-y1)²).
  • Parallel lines have the same gradient, while perpendicular lines have gradients that multiply to -1.
  • A circle with centre (h, k) and radius r has equation (x-h)² + (y-k)² = r².

Using Equations in Coordinate Geometry

  • An equation in two variables can represent a straight line, curve or other figure in the coordinate plane and is called a graph of the equation.
  • Finding the intersect of lines involves solving their equations simultaneously.
  • The equation of a line segment can be found by determining the equation of the line between the two endpoints of the segment and then restricting the domain and range to those endpoints.
  • The equation of a circle can be written in general form or standard form, the latter of which clearly shows the centre and radius.
  • The equation of a parabola can also be determined using different forms, each with its own benefits for different types of analysis.

Remember, much of coordinate geometry involves applying algebraic techniques to geometric ideas, so be sure you are comfortable with both maths topics when studying this area.