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C.P. Algebra IIC.P. Algebra II
The Conic Sections The Conic Sections IndexIndex
The ConicsThe Conics
TranslationsTranslations
Completing the SquareCompleting the Square
Classifying ConicsClassifying Conics
The ConicsThe ConicsParabolParabolaa
CircleCircle
EllipseEllipse
HyperbolHyperbolaa
Click on a Click on a PhotoPhoto
Back to IndexBack to Index
The ParabolaThe Parabola
A parabola is A parabola is formed when formed when
a plane a plane intersects a intersects a cone and the cone and the base of that base of that
conecone
ParabolasParabolas
AA Parabola Parabola is a set of points equidistant is a set of points equidistant from a fixed point and a fixed line.from a fixed point and a fixed line.
The fixed point is called the The fixed point is called the focusfocus..
The fixed line is called the The fixed line is called the directrixdirectrix..
Parabolas Around UsParabolas Around Us
ParabolasParabolas
FOCUSFOCUS
DirectrixDirectrix
ParabolaParabola
Standard form of the Standard form of the equation of a parabola equation of a parabola
with vertex (0,0)with vertex (0,0)
•EquatioEquationn
•FocusFocus•DirectriDirectri
xx•AxisAxis
•xx22=4=4pypy
•(0,p)(0,p)•y = -y = -
pp
•yy22=4=4pxpx
•(p,0)(p,0) •y = py = p
To Find pTo Find p4p is equal to the term in front of 4p is equal to the term in front of x or y. Then solve for p.x or y. Then solve for p.
Example:Example:
xx22=24y=24y
4p=244p=24
p=6p=6
Examples for ParabolasExamples for ParabolasFind the Focus and DirectrixFind the Focus and Directrix
Example 1Example 1
y = 4xy = 4x22
xx22= (= (11//44)y)y
4p = 4p = 11//44
p = p = 11//1616
FOCUSFOCUS
(0, (0, 11//1616))
DirectrixDirectrix
Y = - Y = - 11//1616
Examples for ParabolasExamples for ParabolasFind the Focus and DirectrixFind the Focus and Directrix
Example 2Example 2
x = -3yx = -3y22
yy22= (= (-1-1//33)x)x
4p = 4p = -1-1//33
p = p = -1-1//1212
FOCUSFOCUS
((-1-1//1212, 0), 0)
DirectrixDirectrix
x = x = 11//1212
Examples for ParabolasExamples for ParabolasFind the Focus and DirectrixFind the Focus and Directrix
Example 3 Example 3 (try this (try this one on one on your own)your own)
y = -6xy = -6x22
FOCUSFOCUS
????????
DirectrixDirectrix
????????
Examples for ParabolasExamples for ParabolasFind the Focus and DirectrixFind the Focus and Directrix
Example 3Example 3
y = -6xy = -6x22
FOCUSFOCUS
(0, -(0, -11//2424))
DirectrixDirectrix
y = y = 11//2424
Examples for ParabolasExamples for ParabolasFind the Focus and DirectrixFind the Focus and Directrix
Example 4 Example 4 (try this (try this one on one on your own)your own)
x = 8yx = 8y22
FOCUSFOCUS
????????
DirectrixDirectrix
????????
Examples for ParabolasExamples for ParabolasFind the Focus and DirectrixFind the Focus and Directrix
Example 4Example 4
x = 8yx = 8y22
FOCUSFOCUS
(2, 0)(2, 0)
DirectrixDirectrix
x = -2x = -2
Parabola ExamplesParabola Examples
Now write an equation in Now write an equation in standard form for each of standard form for each of
the following four the following four parabolasparabolas
Write in Standard FormWrite in Standard Form
Example 1Example 1Focus at (-4,0)Focus at (-4,0)
Identify equationIdentify equationyy2 2 =4px p = -4=4px p = -4yy2 2 = 4(-4)x= 4(-4)xyy2 2 = -16x= -16x
Write in Standard FormWrite in Standard Form
Example 2Example 2With directrix y = 6With directrix y = 6Identify equationIdentify equationxx2 2 =4py p = -6=4py p = -6xx2 2 = 4(-6)y= 4(-6)yxx2 2 = -24y= -24y
Write in Standard FormWrite in Standard Form
Example 3 (Now try this Example 3 (Now try this oneone
on your own)on your own)
With directrix x = -1With directrix x = -1
yy2 2 = 4x= 4x
Write in Standard FormWrite in Standard Form
Example 4 (On your own)Example 4 (On your own)
Focus at (0,3)Focus at (0,3)
xx2 2 = 12y= 12y
Back to ConicsBack to Conics
CirclesCircles
A Circle is formed when a A Circle is formed when a plane intersects a cone plane intersects a cone
parallel to the base of the parallel to the base of the cone.cone.
CirclesCircles
Standard Equation of a Standard Equation of a Circle with Center (0,0)Circle with Center (0,0)
x 2 y 2 r 2
Circles & Points of Circles & Points of IntersectionIntersection
Distance formula used Distance formula used to find the radiusto find the radius
(x1 x2 )2 (y1 y2)2 r
CirclesCirclesExample 1Example 1
Write the equation of the circle with the Write the equation of the circle with the point (4,5) on the circle and the origin point (4,5) on the circle and the origin as it’s center.as it’s center.
Example 1Example 1
Point (4,5) on the circle and the origin Point (4,5) on the circle and the origin as it’s center.as it’s center.
(x1 x2 )2 (y1 y2)2 r
(4 0)2 (5 0)2 r
16 25 r
41 r
x 2 y 2 41
Example 2Example 2Find the intersection points on the graph of Find the intersection points on the graph of
the following two equationsthe following two equations
x 2 y 2 25
y 2x 2
x 2 (2x 2)2 25
x 2 4x2 8x 4 25
5x2 8x 4 25
5x2 8x 21 0
(5x 7)(x 3) 0
(5x 7) 0
(x 3) 0
x 3
x 7
5
Now what??!!??!!??Now what??!!??!!??
Example 2Example 2Find the intersection points on the graph of Find the intersection points on the graph of
the following two equationsthe following two equations
x 2 y 2 25
y 2x 2
x 3
x 7
5
Plug these Plug these in for xin for x.
Example 2Example 2Find the intersection points on the graph of Find the intersection points on the graph of
the following two equationsthe following two equations
x 2 y 2 25
y 2x 2
y 2(7
5) 2
y 24
5
7
5,24
5
x 7
5
x 3
y 2( 3) 2
y 4
3, 4 Back to ConicsBack to Conics
EllipsesEllipses
EllipsesEllipses
Examples of EllipsesExamples of Ellipses
EllipsesEllipses
Horizontal Major AxisHorizontal Major Axis
FOCIFOCI(-c,0) & (c,0)(-c,0) & (c,0)
CO-VERTICESCO-VERTICES(0,b)& (0,-b)(0,b)& (0,-b)
CENTER (0,0)CENTER (0,0) VerticesVertices(-a,0) & (a,0)(-a,0) & (a,0)x2
a2y2
b2 1
EllipsesEllipses
Vertical Major AxisVertical Major Axis
FOCIFOCI(0,-c) & (0,c)(0,-c) & (0,c)
CO-VERTICESCO-VERTICES(b, 0)& (-b,0)(b, 0)& (-b,0)
VerticesVertices(0,-a) & (0, a)(0,-a) & (0, a)
CENTER (0,0)CENTER (0,0)
x 2
b2 y 2
a2 1
Ellipse NotesEllipse Notes Length of major axis = a (vertex & Length of major axis = a (vertex &
larger #)larger #)
Length of minor axis = b (co-vertex Length of minor axis = b (co-vertex & smaller#)& smaller#)
To Find the foci (c) use: To Find the foci (c) use:
cc22 = a = a2 -2 - b b22
Ellipse ExamplesEllipse ExamplesFind the Foci and VerticesFind the Foci and Vertices
x2
144y 2
1690
avertices
a13
vertices (0,13),(0, 13)
c 2 a2 b2
c 2 169 144
c 2 25
c 5
foci(0,5),(0, 5)
Ellipse ExamplesEllipse ExamplesFind the Foci and VerticesFind the Foci and Vertices
x 2
81y 2
90
avertices
a9
vertices (9,0),( 9,0)
c 2 a2 b2
c 2 81 9
c 2 72
c 72
foci( 72,0),( 72,0)
Write an equation of an ellipse Write an equation of an ellipse whose vertices are (-5,0) & (5,0) whose vertices are (-5,0) & (5,0) and whose co-vertices are (0,-3) and whose co-vertices are (0,-3)
& (0,3). Then find the foci.& (0,3). Then find the foci.
a5
a2 25
b 3
b2 9
x 2
25y 2
91
c 2 a2 b2
c 2 25 9
c 2 16
c 4
foci(4,0),( 4,0)
Write the equation in standard Write the equation in standard form and then find the foci and form and then find the foci and
vertices.vertices.
49x 2 64 y2 3136
c 2 a2 b2
c 2 64 49
c 2 15
c 15
foci( 15,0),( 15,0)
49x 2
3136
64y2
3136
3136
3136
x 2
64y2
491
vert.(8,0),( 8,0)
Back to the Back to the ConicsConics
The HyperbolaThe Hyperbola
Hyperbola ExamplesHyperbola Examples
Hyperbola NotesHyperbola NotesHorizontal Transverse AxisHorizontal Transverse Axis
Center Center (0,0)(0,0)
Vertices (a,0) Vertices (a,0) && (-a,0)(-a,0)
Foci (c,0) &Foci (c,0) & (-c, 0)(-c, 0)
AsymptotAsymptoteses
Hyperbola NotesHyperbola NotesHorizontal Transverse AxisHorizontal Transverse Axis
EquationEquation
x 2
a2 y2
b2 1
Foci : c2 a2 b2
Hyperbola NotesHyperbola NotesHorizontal Transverse AxisHorizontal Transverse Axis
To find To find asymptotesasymptotes
y b
ax
y bax
Hyperbola NotesHyperbola NotesVertical Transverse AxisVertical Transverse Axis
Center Center (0,0)(0,0)
Vertices (a,0) Vertices (a,0) && (-a,0)(-a,0)
Foci (c,0) &Foci (c,0) & (-c, 0)(-c, 0)
AsymptotAsymptoteses
Hyperbola NotesHyperbola NotesVertical Transverse AxisVertical Transverse Axis
EquationEquation
y 2
a2 x2
b2 1
Foci : c2 a2 b2
Hyperbola NotesHyperbola NotesVertical Transverse AxisVertical Transverse Axis
To find To find asymptotesasymptotes
y a
bx
y abx
Write an equation of the Write an equation of the hyperbola with foci (-5,0) & (5,0) hyperbola with foci (-5,0) & (5,0)
and vertices (-3,0) & (3,0)and vertices (-3,0) & (3,0)
c 2 a2 b2
52 32 b2
25 9 b2
b2 16
x 2
9y2
161
a = 3 c = 5a = 3 c = 5
Write an equation of the Write an equation of the hyperbola with foci (0,-6) & (0,6) hyperbola with foci (0,-6) & (0,6)
and vertices (0,-4) & (0,4)and vertices (0,-4) & (0,4)
c 2 a2 b2
62 42 b2
36 16 b2
b2 20
y 2
16x2
201
a = 4 c = 6a = 4 c = 6
The ConicsThe Conics
TranslationsTranslations
BackBack
What happens when the What happens when the conic is NOT centered on conic is NOT centered on (0,0)?(0,0)?
NextNext
TranslationsTranslationsCircleCircle
NextNext
(x h)2 (y k)2 r 2
TranslationsTranslationsParabolaParabola
NextNext
oror
Horizontal AxisHorizontal Axis
Vertical AxisVertical Axis
(y k)2 4 p(x h)
(x h)2 4 p(y k)
TranslationsTranslationsEllipseEllipse
NextNext
oror
(x h)2
a2 (y k)2
b2 1
(x h)2
b2 (y k)2
a2 1
TranslationsTranslationsHyperbolaHyperbola
NextNext
oror
(x h)2
a2 (y k)2
b2 1
(y k)2
a2 (x h)2
b2 1
TranslationsTranslationsIdentify the conic and graphIdentify the conic and graph
NextNext
(x 1)2 (y 2)2 32
r=r= 33 centercenter (1,-2)(1,-2)
TranslationsTranslationsIdentify the conic and graphIdentify the conic and graph
NextNext
(x 2)2
32 (y 1)2
22 1
TranslationsTranslationsIdentify the conic and graphIdentify the conic and graph
NextNext
(x 3)2
12 (y 2)2
32 1
centecenterrasymptotasymptoteses
verticesvertices
TranslationsTranslationsIdentify the conic and graphIdentify the conic and graph
(x 2)2 4( 1)(y 3)
centecenterr
ConiConicc
Back to IndexBack to Index
Completing the SquareCompleting the Square
Here are the steps for completing the Here are the steps for completing the squaresquare
StepsSteps
1)1) Group xGroup x22 + x, y + x, y22+y move constant+y move constant
2)2) Take # in front of x, ÷2, square, Take # in front of x, ÷2, square, add to both sidesadd to both sides
3)3) Repeat Step 2 for y if neededRepeat Step 2 for y if needed
4)4) Rewrite as perfect square binomialRewrite as perfect square binomialNextNext
Completing the SquareCompleting the Square
Circle: xCircle: x22+y+y22+10x-6y+18=0+10x-6y+18=0
xx22+10x+____ + y+10x+____ + y22-6y=-18-6y=-18
(x(x22+10x+25) + (y+10x+25) + (y22-6y+9)=--6y+9)=-18+25+918+25+9
(x+5)(x+5)22 + (y-3) + (y-3)22=16=16
Center (-5,3)Center (-5,3) Radius = 4Radius = 4 NextNext
Completing the SquareCompleting the Square
Ellipse: xEllipse: x22+4y+4y22+6x-8y+9=0+6x-8y+9=0
xx22+6x+____ + 4y+6x+____ + 4y22-8y+____=-9-8y+____=-9
(x(x22+6x+9) + 4(y+6x+9) + 4(y22-2y+1)=--2y+1)=-9+9+49+9+4
(x+3)(x+3)22 + (y-1) + (y-1)22=4=4
(x 3)2
4
(y 1)2
11
C: (-3,1)C: (-3,1)
a=2, b=1a=2, b=1
IndexIndex
Classifying ConicsClassifying Conics
Classifying ConicsClassifying Conics
Given in General Given in General FormForm
Ax 2 Bxy Cy 2 Dx Ey F 0
NextNext
B2 4AC 0
B2 4AC 0
B2 4AC 0
Classifying ConicsClassifying Conics
Given in General Given in General FormForm
Ax 2 Bxy Cy 2 Dx Ey F 0
ExamplesExamples
B2 4AC 0
B2 4AC 0
B2 4AC 0
Classifying ConicsClassifying Conics
Given in general Given in general form, classify the form, classify the
conicconic
5x2 2y2 20x 4y 24 0
A5
B0
C 2
B2 4AC
02 4(5)(2)
40
EllipsEllipseeNextNext
Classifying ConicsClassifying Conics
Given in general Given in general form, classify the form, classify the
conicconic
y 2 8x 12y 0
A0
B0
C 1
B2 4AC
02 4(0)(1)
0
ParabolParabolaa
NextNext
Classifying ConicsClassifying Conics
Given in general Given in general form, classify the form, classify the
conicconic
24x 2 18y2 18 0
A 24
B18
C 0
B2 4AC
02 4( 24)(18)
1728
HyperboHyperbolala
NextNext
Classifying ConicsClassifying Conics
Given in general Given in general form, classify the form, classify the
conicconic
16 xy0
A0
B 1
C 0
B2 4AC
( 1)2 4(0)(0)
1
HyperboHyperbolala
Back to IndexBack to Index
Classifying ConicsClassifying ConicsGiven in General Given in General FormForm
ThenThen
ORORIf A = CIf A = C
EllipsEllipsee
CirclCirclee
BackBack
Classifying ConicsClassifying ConicsGiven in General Given in General FormForm
ThenThen
BackBack
Classifying ConicsClassifying ConicsGiven in General Given in General FormForm
ThenThen
HyperbolaHyperbola
BackBack