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.