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.

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.

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.

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.

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.