# 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.