Plane Analytical Geometry

By M Bourne

conic section

The curves that we learn about in this chapter are called conic sections. They arise naturally in many situations and are the result of slicing a cone at various angles.

Depending on where we slice our cone, and at what angle, we will either have a straight line, a circle, a parabola, an ellipse or a hyperbola. Of course, we could also get a single point, too.

An interesting application from nature:

The Nautilus Shell


See: Equiangular spiral.

(Image from Tree of Life)

Why study analytic geometry?

Science and engineering involves the study of quantities that change relative to each other (for example, distance-time, velocity-time, population-time, force-distance, etc).

It is much easier to understand what is going on in these problems if we draw graphs showing the relationship between the quantities involved.

The study of calculus depends heavily on a clear understanding of functions, graphs, slopes of curves and shapes of curves. For example, in the Differentiation chapter we use graphs to demonstrate relationships between varying quantities.

We begin with some basic definitions of slope, parallel lines and perpendicular lines.