Conic Sections

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• Conic Sections
• There are 4 types of Conics which we will
  investigate
• Circles
• Parabolas
• Ellipses
• Hyperbolas

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y
(x, y)
r
(h, k)
x
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Graph
by hand.
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y
(-1, 7)
(3,3)
(-5, 3)
(-1,3)
x
(-1, -1)
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Find the center, the radius and graph x2
y2 10x 4y 20 0
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Find the equation of the circle with radius 3 in
QI and tangent to the y-axis at ( 0 , 2 )
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Find the equation of the circle with center at (
2 , -1 ) through ( 5 , 3 )
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Find the equation of the circle with endpoints
of the diameter at ( 3, 5 ) and ( 3 , 1 )
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Find the equation of the circle that Goes
through these 3 points (3, 4), (-1, 2), (0, 3)
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A parabola is defined as the collection of all
points P in the plane that are the same distance
from a fixed point F as they are from a fixed
line D. The point F is called the focus of the
parabola, and the line D is its directrix. As a
result, a parabola is the set of points P for
which
d(F, P) d(P, D)
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The standard form of the equation of a parabola
with directrix parallel to the y-axis is
(opens left or right)
The standard form of the equation of a parabola
with directrix parallel to the x-axis is
(opens up or down)
Where (h, k) represents the vertex of the
parabola and p represents the distance from
the vertex to the focus.
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The Axis of Symmetry is the line through which
the parabola is symmetrical. The Latus Rectum is
a line segment perpendicular to the Axis of
Symmetry through the focus with endpoints on the
parabola. The length of the Latus Rectum is
4p. The Latus Rectum helps define the width
of the parabola.
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focus
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Parabolas Example Problems
Write the equation of a parabola with vertex ( 0
, 0 ) and focus ( 2, 0 )
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Parabolas Example Problems
Find the focus, directrix, vertex, and axis of
symmetry.   y2 12x 2y 25 0  
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Parabolas Example Problems
Write the equation of the parabola with focus (
0 , -2 ) and directrix x 3
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Parabolas Example Problems
Find the focus, directrix, vertex, and axis of
symmetry.   x2 4x 2y 10 0  
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Parabolas Example Problems
Write the equation of the parabola with vertex (
4 , 2 ) and directrix y 5
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Parabolas Example Problems
Write the equation of the parabola with
directrix y 3 and focus ( 3 , 5 )
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Parabolas Example Problems
Find the focus, directrix, vertex, axis of
symmetry, and length of the latus rectum.   x2
4x 12y 32 0  
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Parabolas Example Problems
Write this equation of a parabola in standard
form
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Find the vertex, focus and
directrix of

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Vertex (h, k) (-2, -3)
p 2
Focus (-2, -3 2) (-2, -1)
Directrix y -2 -3 -5
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(-6, -1)
(2, -1)
y -5
(-2, -3)
(-2, -1)
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An ellipse is the collection of points in the
plane the sum of whose distances from two fixed
points, called the foci, is a constant.
y
Minor Axis
P (x, y)
Major Axis
x
Focus2
Focus1
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The standard form of the equation of an ellipse
with major axis parallel to the x-axis is

The standard form of the equation of an ellipse
with major axis parallel to the y-axis is

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(h,k) is the center of the ellipse For any
ellipse, 2a represents the distance along
the major axis (a is always greater than b) 2b
represents the distance along the minor
axis c represents the distance from the
center to either focus (the foci of an ellipse
are always along the major axis)
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Ellipse with Major Axis Parallel to the x-Axis
The ellipse is like a circle, stretched more in
the x direction
y
Focus 2
Focus 1
Major axis
(h, k)
x
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Ellipse with Major Axis Parallel to the y-Axis
The ellipse is like a circle, stretched more in
the y direction
y
Focus 1
(h, k)
Focus 2
x
Major axis
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Ellipses Example Problems
Sketch the ellipse and find the center, foci,
and the length of the major and minor axes
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Ellipses Example Problems
Find the center and the foci. Sketch the
graph.
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Ellipses Example Problems
Write the equation of the ellipse with center (
0 , 0 ), a horizontal major axis, a 6 and b
4
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Ellipses Example Problems
Write the equation of the ellipse with
x-intercepts ? and y-intercepts
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Ellipses Example Problems
Write the equation of the ellipse with foci ( -2
, 0 ) and ( 2 , 0 ), a 7
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Ellipses Example Problems
Write the equation of this ellipse
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Ellipses Example Problems
Find the center, foci, and graph the ellipse
16x2 4y2 96x 8y 84 0
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Ellipses Latus Rectum
The length of the Latus Rectum for an Ellipse is
By knowing the Latus Rectum, it makes the
graph of the ellipse more accurate

Latus Rectum
Latus Rectum
(h, k)
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Ellipses Latus Rectum
Use the length of the latus rectum in Graphing
the following ellipse
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Find an equation of the ellipse with center at
the origin, one focus at (0, 5), and a vertex at
(0, -7). Graph the equation by hand
Center (0, 0)
Major axis is the y-axis, so equation is of the
form
Distance from center to focus is 5, so c 5
Distance from center to vertex is 7, so a 7
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(0, 7)
FOCI
(0, -7)
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Center (h, k) (-4, 2)
Major axis parallel to the x-axis
Vertices (h a, k) (-4 3, 2) or (-7, 2) and
(-1, 2)
Foci (h c, k)
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(-4, 4)
V(-1, 2)
V(-7, 2)
F(-6.2, 2)
F(-1.8, 2)
C (-4, 2)
(-4, 0)
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Hyperbola with Transverse Axis Parallel to the
x-Axis
Latus Rectum
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Hyperbola with Transverse Axis Parallel to the
y-Axis
Latus Rectum
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The length of the Latus Rectum for a Hyperbola is

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A hyperbola is the collection of points in the
plane the difference of whose distances from two
fixed points, called the foci, is a constant.
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The standard form of the equation of a hyperbola
with transverse axis parallel to x-axis is

The standard form of the equation of a hyperbola
with transverse axis parallel to y-axis is

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(h,k) is the center of the hyperbola For any
hyperbola, 2a represents the distance along
the transverse axis 2b represents the distance
along the conjugate axis c represents the
distance from the center to either focus (the
foci of a hyperbola are always along the
transverse axis)
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The equations of the asymptotes for the
hyperbola are these if there is a Horizontal
Transverse Axis or these if there is a
Vertical Transverse Axis
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Hyperbolas Example Problems
Write the equation of the hyperbola with center
( 4 , -2 ) a focus ( 7 , -2 ) and a vertex ( 6,
-2 )
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Hyperbolas Example Problems
Find the center, foci, and graph the hyperbola

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Hyperbolas Example Problems
Find the center, foci, the length of The latus
rectum, and graph the hyperbola
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Hyperbolas Example Problems
Find the center, foci, and vertices
16x2 4y2 96x 8y 76 0
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Equilateral Hyperbolas
Equilateral Hyperbola A hyperbola where a
b. When we have an equilateral hyperbola whose
asymptotes are the coordinate axes, the equation
of the hyperbola looks like this xy k. This
type of hyperbola is called a rectangular
hyperbola, and is easier to graph because the
asymptotes are the x and y axes.  
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Rectangular Hyperbolas
The equation of an rectangular hyperbola is xy
k (where k is a constant value). If k gt0, then
your graph looks like this If klt0, then
your graph looks like this
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Rectangular Hyperbolas
Example Graph by hand the hyperbola xy 6.
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The General form of the equation of any conic
section is
Where A, B, and C are not all zero (however, for
all of the examples we have studied so far, B
0). If A C, then the conic is a If either A
or C is zero, then we have a If A and C have
the same sign, but A does not equal C, then the
conic is a If A and C have opposite signs, then
we have a .
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Conic Sections Eccentricity
Let D denote a fixed line called the directrix
let F denote a fixed point called the focus,
which is not on D and let e be a fixed positive
number called the eccentricity. A conic is the
set of points P in the plane such that the ratio
of the distance from F to P to the distance from
D to P equals e. Thus, a conic is the collection
of points P for which
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Conic Sections Eccentricity
To each conic section (ellipse, parabola,
hyperbola, circle) there is a number called the
eccentricity that uniquely characterizes the
shape of the curve.
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Conic Sections Eccentricity
If e 1, the conic is a parabola. If e 0, the
conic is a circle. If e lt 1, the conic is an
ellipse. If e gt 1, the conic is a hyperbola.
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Conic Sections Eccentricity
For both an ellipse and a hyperbola
where c is the distance from the center to the
focus and a is the distance from the center to a
vertex.
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Conic Sections Eccentricity
Find the eccentricity for the following conic
section 4y2 8y 9x2 54x 49 0
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Conic Sections Eccentricity
Find the eccentricity for the following conic
section 6y2 24y 6x2 12 0
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Conic Sections Eccentricity
Write the equation of the hyperbola with center
( -3 , 1 ) focus ( 2 , 1 ) and e 5/4
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Conic Sections Eccentricity
Write the equation of an ellipse with center ( 0
, 3 ), major axis 12, and eccentricity 2/3
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Conic Sections Eccentricity
Write the equation of the ellipse and find the
eccentricity, given it has foci ( 1 , -1 ) and (
1 , 5 ) and goes through the point ( 4, 2 )
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Conic Sections Eccentricity
Find the center, the foci, and eccentricity.   EX
1 4x2 9y2 36 EX 2 4y2 8y - 9x2
54x 49 0    EX 3 25x2 y2 100x 6y
84 0  
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Conic Sections Solving Systems of Equations
Graphically
Solve the following System of Equations by
Graphing. 9x2 9y2 36 Y 4x 5
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Conic Sections Solving Systems of Equations
Graphically
Solve the following system of equations by
Graphing. x2 -4y 5x2 y2 25
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Conic Sections Solving Systems of Equations
Graphically
Graph the following System, then state a sample
solution.
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Conic Sections Solving Systems of Equations
Graphically
Graph the following System, then state a sample
solution.
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Conics can be formed by the intersection of a
plane with a conical surface. If the plane
passes through the Vertex of the conical
surface, the intersection is a Degenerate Case (a
point, a line, or two intersecting lines).

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Write the equation for this hyperbola
(-2, 8)
(-4, 4)
(0, 4)
C(-2,4)
(-2, 0)
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Theorem Identifying Conics without
Completing the Square
Excluding degenerate cases, the equation
(a) Defines a parabola if AC 0. (b) Defines an
ellipse (or a circle) if AC gt 0. (c) Defines a
hyperbola if AC lt 0.
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Identify the equation without completing the
square.
The equation is a hyperbola.
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Let D denote a fixed line called the directrix
let F denote a fixed point called the focus,
which is not on D and let e be a fixed positive
number called the eccentricity. A conic is the
set of points P in the plane such that the ratio
of the distance from F to P to the distance from
D to P equals e. Thus, a conic is the collection
of points P for which
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If e 1, the conic is a parabola. If e lt 1, the
conic is an ellipse. If e gt 1, the conic is a
hyperbola.
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d(D,P)
r
p
Pole O
(Focus F)
Directrix D
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For both an ellipse and a hyperbola
where c is the distance from the center to the
focus and a is the distance from the center to a
vertex.
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Find an equation of the parabola with vertex at
the origin and focus (-2, 0). Graph the equation
by hand and using a graphing utility.
Vertex (0, 0) Focus (-2, 0) (-a, 0)
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The line segment joining the two points above and
below the focus is called the latus rectum.
Let x -2 (the x-coordinate of the focus)
The points defining the latus rectum are (-2, -4)
and (-2, 4).
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(-2, 4)
(0, 0)
(-2, -4)
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Theorem Equation of an Ellipse Center at (0,
0) Foci at ( c, 0) Major Axis along the
x-Axis
An equation of the ellipse with center at (0,
0) and foci at (- c, 0) and (c, 0) is
The major axis is the x-axis the vertices are at
(-a, 0) and (a, 0).
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y
F2(c, 0)
F1(-c, 0)
(0, b)
x
V1
(-a, 0)
V2(a, 0)
(0, -b)
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Theorem Equation of an Ellipse Center at (0,
0) Foci at (0, c) Major Axis along the
y-Axis
An equation of the ellipse with center at (0,
0) and foci at (0, - c) and (0, c) is
The major axis is the y-axis the vertices are at
(0, -a) and (0, a).
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y
V2 (0, a)
F2 (0, c)
(b, 0)
(-b, 0)
x
F1 (0, -c)
V1 (0, -a)
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Theorem Equation of a Hyberbola Center at (0,
0) Foci at ( c, 0) Vertices at ( a, 0)
Transverse Axis along the x-Axis
An equation of the hyperbola with center at (0,
0), foci at ( - c, 0) and (c, 0), and vertices at
( - a, 0) and (a, 0) is
The transverse axis is the x-axis.
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Theorem Equation of a Hyberbola Center at (0,
0) Foci at ( 0, c) Vertices at (0, a)
Transverse Axis along the y-Axis
An equation of the hyperbola with center at (0,
0), foci at (0, - c) and (0, c), and vertices at
(0, - a) and (0, a) is
The transverse axis is the y-axis.
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Theorem Asymptotes of a Hyperbola
The hyperbola
has the two oblique asymptotes
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Theorem Asymptotes of a Hyperbola
The hyperbola
has the two oblique asymptotes
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Find an equation of a hyperbola with center at
the origin, one focus at (0, 5) and one vertex at
(0, -3). Determine the oblique asymptotes.
Graph the equation by hand and using a graphing
utility.
Center (0, 0)
Focus (0, 5) (0, c)
Vertex (0, -3) (0, -a)
Transverse axis is the y-axis, thus equation is
of the form
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25 - 9 16
Asymptotes
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V (0, 3)
F(0, 5)
(4, 0)
(-4, 0)
F(0, -5)
V (0, -3)
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Find the center, transverse axis, vertices, foci,
and asymptotes of
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Center (h, k) (-2, 4)
Transverse axis parallel to x-axis.
Vertices (h a, k) (-2 2, 4) or (-4, 4)
and (0, 4)
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(h, k) (-2, 4)
Asymptotes
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y - 4 2(x 2)
y - 4 -2(x 2)
(-2, 8)
V (-4, 4)
V (0, 4)
F (2.47, 4)
F (-6.47, 4)
C(-2,4)
(-2, 0)
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Parabola with Axis of Symmetry Parallel to
x-Axis, Opens to the Right, a gt 0.
D x -a h
y
V (h, k)
Axis of symmetryy k
F (h a, k)
x
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Parabola with Axis of Symmetry Parallel to
x-Axis, Opens to the Left, a gt 0.
D x a h
y
Axis of symmetry y k
F (h - a, k)
x
V (h, k)
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Parabola with Axis of Symmetry Parallel to
y-Axis, Opens up, a gt 0.
Axis of symmetry x h
y
F (h, k a)
V (h, k)
D y - a k
x
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Parabola with Axis of Symmetry Parallel to
y-Axis, Opens down, a gt 0.
Axis of symmetry x h
y
D y a k
V (h, k)
F (h, k - a)
x
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The standard form of an equation of a circle of
radius r with center at the origin (0, 0) is
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Graph
using a
graphing utility.
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Parabolas Example Problems
The LATUS RECTUM is a line segment, either
horizontal or vertical, that joins the focus with
two points on the parabola. The length of the
Latus Rectum 4a.
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Latus Rectum Let y -1
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Ellipse with Major Axis Parallel to the x-Axis
where a gt b and b2 a2 - c2.
y
(h c, k)
(h - c, k)
Major axis
(h - a, k)
(h a, k)
(h, k)
x
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Ellipse with Major Axis Parallel to the y-Axis
where a gt b and b2 a2 - c2.
y
(h, k a)
(h, k c)
(h, k)
(h, k - c)
x
Major axis
(h, k - a)
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Hyperbola with Transverse Axis Parallel to the
y-Axis Center at (h, k) where b2 c2 - a2.
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Hyperbola with Transverse Axis Parallel to the
x-Axis Center at (h, k) where b2 c2 - a2.
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Conic Sections Part I CIRCLES
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y
D x -a
V
x
F (a, 0)
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y
D x a
V
x
F (-a, 0)
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y
F (0, a)
x
V
D y -a
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y
D y a
x
F (0, -a)
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Lesson Overview 10-8A
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Lesson Overview 10-8B