Introduction

Analytical (coordinate) geometry studies geometric figures using algebra, by fixing a coordinate system and translating geometric conditions into equations. The idea was born in the 1630s, when two French mathematician-philosophers, René Descartes (1596–1650) and Pierre de Fermat, independently founded the subject by adapting François Viète's (1540–1603) newly systematic algebraic notation to the study of geometric loci. Descartes described his coordinate system as 'a device to locate points on a plane' and analytical geometry as 'a way of visualising algebraic formulas' — bridging algebra and geometry for the first time. Because of Descartes' role, analytical geometry is also called Cartesian geometry.

From the 17th century, mathematics split into pure and applied strands, and one of the very first applied topics studied was straight-line motion. Straight-line graphs turn out to model an enormous range of real situations — business and economics, the social sciences, physics, medicine — precisely because many real quantities change at a constant rate, which is exactly what a straight line represents. Solving a real-world problem starts by formulating it mathematically: identifying the two quantities involved, choosing which is the independent variable ($x$-axis) and which the dependent variable ($y$-axis), and constructing the linear model from the given data.

The chapter is framed around several such motivating situations — worked out later using the tools developed here: a student's walking speed and school start-time (linear relation between speed and time-of-arrival, leading to a pair of straight lines describing the two walking scenarios); a power company choosing where along a road to build a substation serving two villages (shortest total cable length, via reflection and the two-points/family-of-lines ideas); an ant on the outside of a cylindrical vessel finding the shortest crawl to a honey drop on the inside (solved by unrolling the curved surface into a flat rectangle, turning a 3-D shortest-path problem into an ordinary straight-line distance); and a compact-disk manufacturer's linear demand and supply curves (finding the market equilibrium price and quantity as the intersection of two lines).

By the end of this chapter you should be able to: write the equation of a line in each of its standard forms; decide whether two given lines are parallel or perpendicular; find the distance of a point from a line and the distance between two parallel lines; describe a family of straight lines satisfying a given condition; and write down the equation of a pair of straight lines, the angle between them, and their angle bisectors.