Transcript CHAPTER 7

CHAPTER 9
CONIC SECTIONS
9.1 THE ELLIPSE
• Objectives
– Graph ellipses centered at the origin
– Write equations of ellipses in standard form
– Graph ellipses not centered at the origin
– Solve applied problems involving ellipses
Definition of an ellipse
• All points in a plane the sum of whose distances
from 2 fixed points (foci) is constant.
• If an ellipse has a center at the origin and the
horizontal axis is 2a (distance from center to right
end is a) and the vertical axis is 2b (distance from
the center to the top is b), the equation of the
ellipse is:
x
2
a
2

y
2
b
2
1
Graph
x
2
9

y
2
1
16
• Center is at the origin, horizontal axis=6 (left
endpt (-3,0), right endpt (3,0), vertical axis = 8
(top endpt (0,4), bottom endpt (0,-4))
What is c?
• c is the distance from the center to the focal point
c  a b
2
2
2
What is the equation of an ellipse,
centered at the origin with a
horizontal axis=10 and vertical
2
2
axis=8?
x
y

1)
1
25
2)
x
16
2

10
3)
x
4)
2

y
2
1
64
2
5
1
8
100
x
y
2

y
2
4
1
What if the ellipse is not centered
at the origin?
• If it is centered at any point, (h,k), the ellipse is
translated. It is moved right “h” units and up “k” units
from the origin.
2
• Consider:
( x  2)
2
 ( y  3)  1
4
• The center is at (2,-3), the distance from the center to
the right & left endpt = 2, the distance to the top &
bottom endpt = 1
• Graph on next slide
• Since a>b, the horizontal axis will be the major axis
and the focal points will be on that axis of the ellipse.
Graph
What is the distance
from the center to c 2
each focal point?(c)
 a b
c
If the center is at
(2,-3), the foci are at
Since the major axis is
the horizontal one,
you move c units left &
right of the center.
(2 
2
2
 4 1  3
3
3 ,  3 ), ( 2 
3 , 3)
9.2 The Hyperbola
• Objectives
– Locate a hyperbola’s vertices & foci
– Write equations of hyperbolas in standard
form
– Graph hyperbolas centered at the origin
– Graph hyperbolas not centered at the origin
– Solve applied problems involving hyperbolas
Definition of a hyperbola
• The set of all points in a plane such that the
difference of the distances to 2 fixed points (foci)
is constant.
• Standard form of a hyperbola centered at the
2
2
origin:
x
y


1
2
2
• Opens left & right
a
b
• Opens up & down y
2
a
2

x
2
b
2
1
What do a & b represent?
• a is the distance to the vertices of the
hyperbola from the center (along the
transverse axis)
• b is the distance from the center along the
non-transverse axis that determines the
spread of the hyperbola (Make a rectangle
around the center, 2a x 2b, and draw 2
diagonals through the box. The 2
diagonals form the oblique asymptotes for
the hyperbola.)
Focal Points
• The foci (focal points) are located “inside” the 2
branches of the hyperbola.
• The distance from the center of the hyperbola to
the focal point is “c”.
c
a b
2
2
• Move “c” units along the transverse axis (vertical
or horizontal) to locate the foci.
• The transverse axis does NOT depend on the
magnitude of a & b (as with the ellipse), rather as
to which term is positive.
Describe the
ellipse:
•
•
•
•
1)
2)
3)
4)
4 x  25 y  100
2
2
opens horizontal, vertices at (4,0),(-4,0)
opens vertically, vertices at (0,5),(0,-5)
opens vertically, vertices at (0,4),(0,-4)
opens horizontal, vertices at (5,0),(-5,0)
What if the hyperbola is not
centered at the origin? (translated)
• A hyperbola with a horizontal transverse
axis, centered at (h,k) is of the form:
( x  h)
a
2
2

(y  k)
b
2
2
1
Describe the hyperbola & graph
9 ( y  3 )  4 ( x  1)  36
2
2
• Transverse axis is vertical
• Centered at (-1,3)
• Distance to vertices from center= 2 units (up &
down) (-1,5) & (-1,1)
• Asymptotes pass through the (-1,3) with slopes =
2/3, -2/3
• Foci 13 units up & down from the center ,
(  1,3 
13 ), (  1,3  13 )
Graph of example
from previous slide
9.3 The Parabola
• Objectives
– Graph parabolas with vertices at the origin
– Write equations of parabolas in standard form
– Graph parabolas with vertices not at the
origin
– Solve applied problems involving parabolas
Definition of a parabola
• Set of all points in a plane equidistant from
a fixed line (directrix) and fixed point
(focus), that is not on the line.
• Recall, we have previously worked with
parabolas. The graph of a quadratic
equation is that of a parabola.
Standard form of a parabola
centered at the origin, p = distance
from the center to the focus
• Opens left (p<0),or right (p>0)
y  4 px
2
• Opens up (p>0) or down (p<0)
x  4 py
2
• Distance from vertix to directrix = -p
Graph and describe y  2 x
• Write in standard form: x 
2
1
y
2
• (1/2)y = (4p)y, thus ½ = 4p, p = 1/8
• Center (0,0), opens up, focus at (0,1/8)
• Directrix: y = -1/8
2
Translate the parabola: center at
(h,k)
• Vertical axis of symmetry
( x  h)  4 p( y  k )
2
• Horizontal axis of symmetry
( y  k )  4 p( x  h)
2
If the equation is not in standard
form, you may need to complete
the square to achieve standard
form.
• Find the vertex, focus, directrix & graph
x  4 x  8 y   12
2
x  4 x   8 y  12
2
( x  2 )   8 y  12  4   8 y  8   8 ( y  1)
2
• Vertex (-2,-1), p= -2, focus: (-2,-3), directrix: y=1
• Graph, next slide
Graph of previous slide example
9.4 Rotation of Axes
• Objectives
– Identify conics without completing the
square
– Use rotation of axes formulas
– Write equations of rotated conics in
standard form
– Identify conics without rotating axes
Identifying a conic without completing
the square (A,C not equal zero)
Ax  Cy  Dx  Ey  F  0
2
•
•
•
•
2
Circle, if A=C
Parabola, if AC=0
Ellipse if AC Not equal 0, AC>0
Hyperbola, if AC<0
A Rotated Conic Section
• Can the graph of a conic be rotated from
the standard xy-coordinate system?
• YES! How do we know when we have a
rotation? When there is an xy term in the
general equation of a conic:
Ax  Bxy  Cy  Dx  Ey  F  0
2
2
Rotation of Axes
• A conic could be rotated through an angle
• The xy-coordinate system is the standard
coordinate system. The x’y’-coordinate system
is the rotated system (turning the rotated conic
into the standard system)
• Coordinates between (x,y) and (x’,y’) for every
point are found according to this relationship:
x  x ' cos   y ' sin 
y  x ' sin   y ' cos 
Expressing equation in standard
form, given a rotated axis.
• Given the equations relating (x,y) and
(x’,y’), find (x’,y’) given the angle of
rotation
• Substitute these expressions in for x in the
equation of the rotated conic. The result is
an equation (in terms of x’ & y’) that exists
IF the equation were in standard position.
How do we determine the amount
of rotation of the axes?
cot 2 
AC
B
Identifying a conic section w/o a
rotation of axes
Parabola : B  4 AC  0
2
Ellipse / circle : B  4 AC  0
2
Hyperbola
: B  4 AC  0
2
9.5 Parametric Equations
• Objectives
– Use point plotting to graph plane curves
described by parametric equations
– Eliminate the parameter
– Find parametric equations for functions
– Understand the advantages of
parametric representations
Plane Curves & Parametric
Equations
• Parametric equations: x & y are defined in
terms of a 3rd variable, t: f(t)=x, g(t)=y
• Various values can be substituted in for t to
produce new values for x & y
• These values can be plotted on the xycoordinate system to generate a graph of the
functions
Given a function in terms of x & y,
can you find its representation as
parametric equations?
• Begin by allowing one variable (usually x)
to designated as t. Replace x with t in the
expression. Now y is stated in terms of t.
• x may be replace with other expressions
involving t. The only restriction is that x
and the new expression for x (in terms of t)
must have the same domain.
9.6 Conic Sections in Polar
Coordinates
• Objectives
– Define conics in terms of a focus and a
directrix
– Graph the polar equations of conics
Focus-Directrix Definitions of the
Conic Sections
f : focus , D : directrix , P : po int s
e
PF
 eccentrici ty
PD
e  1, parabola
e  1, ellipse
e  1, hyperbola
Polar Equations of Conics
• (r,theta) is a point on the graph of the conic
• e is the eccentricity
• p is the distance between the focus & the
directrix
r 
ep
1  e cos 
,r 
ep
1  e sin 