# Plane Analytical Geometry

By M Bourne

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:

## 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.

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Chapter Contents

- Plane Analytical Geometry
- 1. Distance Formula
- 1a. Gradient (Slope) of a Line, and Inclination
- 1b. Parallel Lines
- 1c. Perpendicular Lines
- 2. The Straight Line
- Perpendicular Distance from a Point to a Line
- 3. The Circle
- 4. The Parabola
- 4a. Interactive parabola graphs
- 5. The Ellipse
- 5a. Interactive ellipse graphs
- 6. The Hyperbola
- 6a. Interactive hyperbola graphs
- 6b. Conic Sections Summary
- 6c. Conic Sections 3D interactive graph
- 7. Polar Coordinates
- 8. Curves in Polar Coordinates
- Equi-angular Spiral
- Squaring the Circle: rope method with proof