Conic Sections

The Parabola

Introduction

• Consider a cone being intersected with a plane

Note the different shaped curves that result

Note the different shaped curves that result

Introduction

• We will consider various conic sections and how they are described analytically

Parabolas Hyperbolas Ellipses Circles

They can be described or

defined as a set of points which satisfy certain

conditions

They can be described or

defined as a set of points which satisfy certain

conditions

Parabola

• Definition A set of points on the plane that are equidistant

from A fixed line

(the directrix) and A fixed point

(the focus) not on the directrix

Parabola

• Note the line through the focus, perpendicular to the directrix Axis of symmetry

• Note the point midway between the directrix and the focus Vertex

View Geogebra DemonstrationView Geogebra Demonstration

Equation of Parabola

• Let the vertex be at (0, 0) Axis of symmetry be y-axis Directrix be the line y = -p (where p > 0) Focus is then at (0, p)

• For any point (x, y) on the parabola

Distance = y + pDistance = 2 20x y p ( , )x y

Equation of Parabola

• Setting the two distances equal to each other

• What happens if p < 0?

• What happens if we have

2 2

2

0

4

x y p y p

x p y

. . . simplifying . . .

2 4 ?y p x

Working with the Formula

• Given the equation of a parabola y = ½ x2

• Determine The directrix The focus

• Given the focus at (-3,0) and the fact that the vertex is at the origin

• Determine the equation

When the Vertex Is (h, k)

• Standard form of equation for vertical axis of symmetry

• Consider What are the coordinates

of the focus? What is the equation

of the directrix?

24x h p y k

(h, k)

When the Vertex Is (h, k)

• Standard form of equation for horizontal axis of symmetry

• Consider What are the coordinates

of the focus? What is the equation

of the directrix?

24y k p x h

(h, k)

Try It Out

• Given the equations below, What is the focus? What is the directrix?

2( 3) ( 2)x y

2 4 9 0x y y

24 12 12 7 0x x y

Another Concept

• Given the directrix at x = -1 and focus at (3,2)

• Determine the standard form of the parabola

Applications

• Reflections of light rays Parallel rays

strike surfaceof parabola

Reflected backto the focus

View Animated DemoView Animated Demo

Build a working parabolic cooker

Build a working parabolic cooker

How to Find the FocusHow to Find the Focus

Proof of the Reflection Property

Proof of the Reflection Property

Spreadsheet DemoSpreadsheet Demo

MIT & Myth BustersMIT & Myth Busters

Applications

• Light rays leaving the focus reflectout in parallel rays

Used for Searchlights

Used for Searchlights

Military Searchlights

Military Searchlights

Assignment

• See Handout

• Part A 1 – 33 odd

• Part B 35 – 43 all