Coordinate geometry

Coordinate Geometry Basics

  • The Cartesian coordinate system is an essential part of coordinate geometry, consisting of horizontal x-axis and vertical y-axis intersecting at a point called the origin denoted as (0,0).
  • A point in the cartesian plane is identified by an ordered pair of coordinates (x, y), where x represents its distance from the y-axis and y represents its distance from the x-axis.
  • The plane is divided into four quadrants by the x and y-axes.

Equation of a Line

  • The equation of a line in two-dimensional space can be represented in different forms: slope-intercept form: y = mx + c, general form: Ax + By + C = 0, and point-slope form: y - y1 = m(x - x1), where m represents the slope and c is the y-intercept.
  • Slope of a line can be calculated using two points on the line: m = (y2 - y1) / (x2 - x1).
  • Horizontal lines have a slope of 0, while vertical lines have an undefined slope.
  • Parallel lines have the same slope, and perpendicular lines have slopes that are negative reciprocals of each other.
  • The y-Intercept of the line is the value of y when x = 0. Any point on the y-axis will have an x coordinate of 0.
  • Midpoint of a line joining two points (x1, y1) and (x2, y2) can be found by the formula: [(x1+x2)/2, (y1+y2)/2].

Equation of a Circle

  • The standard equation of a circle with centre at (h, k) and radius r is: (x - h)^2 + (y - k)^2 = r^2.
  • A circle centered at the origin (0, 0) has the equation: x^2 + y^2 = r^2.

Distance Formula

  • The distance between two points (x1, y1) and (x2, y2) can be found using the formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).

Locus

  • The locus of a point refers to the path traced out by the point when it moves according to a given rule or condition.
  • Common conditions include the point moving at a constant distance from a fixed point (the locus is a circle) or along a straight path (the locus is a straight line).

General Properties

  • The symmetry properties of geometrical figures such as lines, circles can be analyzed from their algebraic representations.
  • The principles of coordinate geometry can be extended to three dimensional spaces, expanding the study to include 3-dimensional figures like spheres and lines.

Vector Geometry

  • Unlike normal numbers, vectors have both magnitude (their length) and direction
  • Mathematical operations can be performed with vectors, such as addition, subtraction, and scalar multiplication.
  • The dot product and cross product are operations that apply to vectors in three dimensions.
  • Unit vectors’ length is 1. They are often used to represent directions.
  • Vectors can be represented through components: a magnitude for each direction (i, j, and k correspond to the X, Y, and Z directions in 3D space, respectively).

Remember that regular practice is key in mastering coordinate geometry. Continue to work through problems and apply the formulas and principles discussed here.