Conic Sections The Parabola Introduction • Consider a cone being intersected with a plane Note the different shaped curves that result.
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Transcript Conic Sections The Parabola Introduction • Consider a cone being intersected with a plane Note the different shaped curves that result.
Conic Sections
The Parabola
Introduction
• Consider a cone being intersected with a
plane
Note the different
shaped curves that result
Introduction
They can be
described or
defined as a set
of points which
satisfy certain
conditions
• We will consider various conic sections and
how they are described analytically
Parabolas
Hyperbolas
Ellipses
Circles
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
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 =
x 0 y p
2
2
( x, y )
Distance = y + p
Equation of Parabola
• Setting the two distances equal to each other
x 0 y p
2
2
y p
. . . simplifying . . .
x2 4 p y
• What happens if p < 0?
2
• What happens if we have y 4 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
2
symmetry
x h 4 p y k
• Consider
What are the coordinates
of the focus?
What is the equation
of the directrix?
(h, k)
When the Vertex Is (h, k)
• Standard form of equation for horizontal axis
2
of symmetry
y k 4 p x h
• Consider
What are the coordinates
of the focus?
What is the equation
of the directrix?
(h, k)
Try It Out
• Given the equations below,
What is the focus?
What is the directrix?
4x 12x 12 y 7 0
2
( x 3) ( y 2)
2
x y 4y 9 0
2
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 surface
of parabola
Reflected back
to the focus
View Animated Demo
Spreadsheet Demo
How to Find the Focus
Proof of the
Reflection Property
Build a working
parabolic cooker
MIT & Myth Busters
Applications
• Light rays leaving the focus reflect
out in parallel rays
Used for
Searchlights
Military
Searchlights
Assignment
• See Handout
• Part A 1 – 33 odd
• Part B 35 – 43 all