# Year 10+ Plane Geometry

### Chapters

### Parabolas

# Parabolas

When you throw a cricket ball, shoot an arrow, kick a footy ball, fire a catapult, fire an arrow or hit a golf ball, it takes an arc-shaped
path through the air and comes down to land again some distance away from you. The path of your `projectile`

is like a `parabola`

, except that
air resistance modifies it a bit.

Telescopes and headlights use reflective lenses shaped like parabolas because special properties of the parabola focus all the light at one point.

So, what exactly is a parabola?

## Definition of the Parabola

Formally, a `parabola`

is the locus of a point that moves so that its distances from a fixed
point called the `focus`

and a fixed line called the `directrix`

are equal

The ratio between the distances of a point to the focus and the distances of a point to the directrix in a conic section is called the `eccentricity`

, and it measures how far the conic
is from being a circle. In a parabola, the eccentricity is always equal to \(1\) because these distances are always equal.

## Parts of a Parabola

When you are doing problems on parabolas, it will help if you can name the important parts of a parabola. They are:

- The
`directrix`

and`focus`

: the fixed line and fixed point talked about above. - The
`axis of symmetry`

: a line through the focus at right angles to the directrix that cuts the parabola evenly in half. - The
`vertex`

: the point where the parabola turns around and goes in the other direction. It is halfway between the focus and the directrix.

## What Makes the Parabola so Useful in Torches and Telescopes?

The parabola has a special property that makes it a great shape to use in

- telescopes
- torches and spot lights
- radar dishes
- satellite dishes
- bananas (not really - but see later)
- solar collectors for solar panels

It's this:

If the inside of a parabola is hit by a ray that is parallel to the parabola's axis of symmetry, then the ray is **reflected**
off the surface of the parabola, directly **to the focus**.

This helps to concentrate light, radio or digital signals for collection at the focus of the parabola.

## The Parabola as a Conic Section

A parabola appears as a cross-section of a cone, when you take a slice parallel to the cone's side.
Thus, the parabola is a `conic section`

. You can read more about conic sections in the article on conic sections.

### The Equation of a Parabola

We can plot parabolas in the \(xy\)-plane. The simplest parabolas have their vertices at the origin, and either face upwards or to the side like these two graphs of \(y = x^2\) and \(y^2 = x\):

The general equation of a sideways parabola (opening to the right) with its vertex at the origin is

Here \(a\) is called the `focal length`

. It is the distance from the vertex to the focus (and from the vertex to the directrix).

We can find equations for a parabola with vertex \((0,0)\) that opens upwards, downwards to the left and to the right as follows:

### Examples

- Find the focus of the parabola \(32y = x^2\). Which way does the parabola face?
**Solution:**The focal length is \(a = 32 \div 4 = 8\). As the equation includes a term in \(x^2\) and the coefficient of \(y\) is positive, the parabola opens upwards. The focus is the point \((0,8)\).

- Find the focus of the parabola \(y^2 = 16 x\). Which way does the parabola face?
**Solution:**The focal length is \(a = 16 \div 4 = 4\). As the equation includes a term in \(y^2\) and the coefficient of \(x\) is positive, the parabola opens to the right. The focus is the point \((4,0)\).

### Latus Rectum of a Parabola

The `latus rectum`

of a hyperbola is a line segment that runs through the focus of the parabola, parallel to the directrix.

### Parabolas in Real Life

Parabolas occur in many situations, both naturally and in art and architecture.

The curve of the banana on the left is a parabola.

The view of the Sydney Harbour Bridge shown on the right includes two different parabolas. Can you find them? What would their equations look like?

### Description

In these chapters you will learn about plane geometry topics such as

- Area (Irregular polygons, plane shapes etc)
- Perimeter
- Conic sections (Circle, Ellipse, Hyperbola etc)
- Polygons (Congruent, polygons, similar, triangles etc)
- Transformations and symmetry (Reflection, symmetry, transformations etc)

etc

Even though these chapters are marked for Year 10 or higher students, several topics are for students in Year 8 or higher

### Audience

Year 10 or higher, suitable for Year 8 + students as well.

### Learning Objectives

Learn about Plane Geometry

Author: Subject Coach

Added on: 28th Sep 2018

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