# Plane Analytical Geometry

By M Bourne

### An interesting application from nature:

The Nautilus Shell

See: Equiangular spiral.

(Image from Tree of Life)

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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.

## 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.

## In this Chapter

- 1. Distance Formula
- Gradient (Slope) of a Line, and Inclination
- Parallel Lines
- Perpendicular Lines
- 2. The Straight Line
- Perpendicular Distance from a Point to a Line
- 3. The Circle
- 4. The Parabola
- 5. The Ellipse
- 6. The Hyperbola
- 7. Polar Coordinates
- 8. Curves in Polar Coordinates
- Equi-angular Spiral

We begin with the Distance Formula »

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