Conic Sections
The Parabola
1
Introduction
• Consider a cone being intersected with a
plane
Note the different
shaped curves that result
2
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
3
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
4
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
5
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
6
Equation of Parabola
• Setting the two distances equal to each other
 x  0   y  p 
2
2
 y p
. . . simplifying . . .
x2  4  p  y
Link to web example
• What happens if p < 0?
2
• What happens if we have y  4  p  x
?
7
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
8
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)
9
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)
10
Try It Out
• Given the equations below,


What is the focus?
What is the directrix?
4 x2  12 x  12 y  7  0
( x  3)  ( y  2)
2
x  y  4y  9  0
2
11
Another Concept
• Given the directrix at x = -1 and focus at (3,2)
• Determine the standard form of the parabola
12
Applications
• Reflections of light rays


Parallel rays
strike surface
of parabola
Reflected back
to the focus
View Animated Demo
How to Find the Focus
Proof of the
Reflection Property
Build a working
parabolic cooker
13
Applications
• Light rays leaving the focus reflect
out in parallel rays
Used for
Searchlights
Military
Searchlights
14
Assignment
• See Handout
• Part A 1 – 33 odd
• Part B 35 – 43 all
15