The Coordinate Geometry of Straight Lines

The Coordinate Geometry of Straight Lines

Core Concepts

  • The equation of a straight line is often expressed in the form y=mx+c, where m is the slope and c is the y-intercept.
  • Gradient, or slope, can be calculated using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two different points on the line.
  • Vertical lines have an undefined or infinite gradient, and their equation is of the form x=a, where a is the x-coordinate of all points on the line.
  • Horizontal lines have a gradient of zero, and their equation is y=b, where b is the y-coordinate of all points on the line.

Intercepts & Intersection Points

  • The y-intercept is the value of y when x is zero; it is represented by c in the equation y=mx+c.
  • An x-intercept is a point where the line crosses the x-axis (where y=0).
  • The coordinates of the intersection of two lines can be found by equating their equations and solving for x and y.

Perpendicular & Parallel Lines

  • Two lines are perpendicular if the product of their gradients is -1; this means the gradient of a line perpendicular to a given line is the negative reciprocal of the initial line’s gradient.
  • Two lines are parallel if they have the same gradient, even if they do not intersect.
  • The distance between parallel lines can be found using the formula d = c2-c1 / √(m^2+1), where m is the common gradient and c1 and c2 are the y-intercepts of the lines.

Equations of Lines

  • Point-slope form allows you to find the equation of a line given the slope and a single point on the line: y - y1 = m(x - x1).
  • Two-point form is used to find the equation of a line given two points on the line: (y - y1) = ((y2 - y1) / (x2 - x1)) * (x - x1).
  • Intercept form expresses the equation of a line using the x and y intercepts: x/a + y/b = 1.

Applications of Coordinate Geometry

  • Coordinate geometry can help solve real-world problems, such as finding the shortest distance between points, designing optimal routes, and much more.
  • Intersection of lines has applications in physics and computer graphics, where understanding the point of intersection is critical to problem-solving.