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04/19/23 Math 120 - KM 1
Chapter 9: Conic Sections
• 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle
• 9.2 Ellipse
• 9.3 Hyperbola
• 9.4 Nonlinear Systems
CH 9 KM & PP AIM2 2
Sections of a Cone
CH 9 KM & PP AIM2 3
Sections of a Cone ... continued
CH 9 KM & PP AIM2 4
Degenerate Conic Sections
04/19/23 Math 120 - KM 5
9.1
04/19/23 Math 120 - KM 6
The Parabola
9.1
04/19/23 Math 120 - KM 7
A Parabolic ReflectorFor a Microphone
Can You Hear a Pin Drop?
9.1
04/19/23 Math 120 - KM 8
A Parabolic Archway
Architectural Parabola
9.1
04/19/23 Math 120 - KM 9
A Parabolic Headlight
Shine Your Light Forward
9.1
04/19/23 Math 120 - KM 10
Parabolic Shadows
9.1
04/19/23 Math 120 - KM 11
y = ax2 + bx + ca > 0 a < 0
x = ay2 + by + c a > 0 a < 0
9.1 The Basic Ideas
9.1
04/19/23 Math 120 - KM 12
{-4,2} {5/3} {17.5327} {3}
542 2 xxy
85442 2 xxy
322 2 xy
Vertex: (-2, -3)
Opens upwards (narrow)
Axis of symmetry: x = -2
y -intercept: (0,5)
9.1 Ex 1: y = 2x2 + 8x + 5
9.1
04/19/23 Math 120 - KM 13
{-4,2} {5/3} {17.5327} {3}
,ab
:vertex2
Vertex: (-2, -3)
Opens upwards (narrow)
Axis of symmetry: x = -2
y -intercept: (0,5)
Upward:Opens
,:erceptinty 0
9.1 Ex 1: y = 2x2 + 8x + 5alternate method
9.1
04/19/23 Math 120 - KM 14
{-4,2} {5/3} {17.5327} {3}
2x2x6y 2
62126 2 xxy
416 2 xyVertex: (1, 4)
Opens downward (narrow)
Axis of symmetry: x = 1
y -intercept: (0,-2)
9.1 Ex 2: y = -6x2 + 12x - 2
9.1
04/19/23 Math 120 - KM 15
Vertex: (1, 4)
Opens downward (narrow)
Axis of symmetry: x = 1
y -intercept: (0-2)
,ab
:vertex2
Downward:Opens
,:erceptinty 0
9.1 Ex 2: y = -6x2 + 12x – 2
alternate method
9.1
04/19/23 Math 120 - KM 16
542 2 yyx
85442 2 yyx
322 2 yx
Vertex: (-3, 2)
Opens to the right (narrow)
Axis of symmetry: y = 2
x – intercept: (5, 0)
9.1 Ex 3: x = 2y2 – 8y + 5
9.1
04/19/23 Math 120 - KM 17
Vertex: (-3, 2)
Opens to the right (narrow)
Axis of symmetry: y = 2
x – intercept: (5, 0)
righttheto:Opens
0,:erceptintx
ab
,:vertex2
9.1 Ex 3: x = 2y2 – 8y + 5alternate method
9.1
04/19/23 Math 120 - KM 18
3y2y2x 2
231y2y2x 2
11y2x 2 Vertex: (-1, -1)
Opens to the left (narrow)
Axis of symmetry: y = -1
x – intercept: (-3, 0)
9.1 Ex 4: x = -2y2 – 4y - 3
9.1
04/19/23 Math 120 - KM 19
Vertex: (-1, -1)
Opens to the left (narrow)
Axis of symmetry: y = -1
x – intercept: (-3, 0)
ab
,:vertex2
lefttheto:Opens
0,:erceptintx
9.1 Ex 4: x = -2y2 – 4y – 3alternate method
9.1
04/19/23 Math 120 - KM 20
111 ,yxP
222 ,yxP
c
a
b
2212
21 yyxxd
22 bac
The Distance Formula
9.1
04/19/23 Math 120 - KM 21
2212
21 yyxxd
Determine the distance
from P1 to P2.
P1 (-2, 3) P2(2, 0)
P1 (5, -2) P2(-3, -1)
9.1 Distance Formula Examples
9.1
04/19/23 Math 120 - KM 22
9.1 MIDPOINT
9.1
04/19/23 Math 120 - KM 23
222 ,yxP
mm yxM ,
2
,2
),( 2121 yyxxyx mm
111 ,yxP
AVERAGE !
9.1 Average the
Coordinates!
9.1
04/19/23 Math 120 - KM 24
Determine the midpoint of
P1P2.
P1 (-2, 3) P2(2, 0)
P1 (5, -2) P2(-3, -1)
9.1 Midpoint Examples
9.1
04/19/23 18:05 krm 11.2 25
With a COMPASS
How do I make a
circle ?
9.1 Circles
9.1
04/19/23 18:05 krm 11.2 26
The set of all points in a plane that are at a fixed distance, r, called the radius from a fixed point, (h, k), called the center. 222 r)ky()hx(
9.1 Circle: Center (h,k) Radius r
9.1
04/19/23 Math 120 - KM 27
9.1 x2 + y2 = 1
9.1
04/19/23 Math 120 - KM 28
9.1 (x + 2)2 + (y – 4)2 = 32
9.1
04/19/23 Math 120 - KM 29
9.1 x2 + (y + 4)2 = 25
9.1
04/19/23 18:05 krm 11.2 30
222 r)ky()hx(
5h 8k 7r
4985 22 )y()x(
9.1 Write the equation of the circle with
radius 7 and center (-5, 8).
222 7)8y()5x( 9.1
04/19/23 18:05 krm 11.2 31
Look forax2 + ay2
How do I know it’s a
circle ?
The Equation of a Circle
9.1
04/19/23 18:05 krm 11.2 32
Write the equation of the circle in standard form and sketch the graph:
x2 + y2 - 6x + 10y + 25 = 0
Circle: Standard Form
25106 22 yyxx
25925251096 22 yyxx
953 22 yx
222 353 yx
9.1
04/19/23 Math 120 - KM 33
9.2 The Ellipse
9.2
04/19/23 Math 120 - KM 34
12
2
2
2
by
ax
x-intercepts (+ a, 0)
y-intercepts (0, + b)
9.2 Ellipse (it fits in a box!)
9.2
04/19/23 Math 120 - KM 35
14
y
16
x 22
1
24 2
2
2
2
yx
),( 04),( 04
),( 20
),( 20
9.2 Example: Horizontal Major Axis
),( 00
9.2
04/19/23 Math 120 - KM 36
116
y
4
x 22
1
42 2
2
2
2
yx
),( 02),( 02
),( 40
),( 40
9.2 Example: Vertical Major Axis
),( 00
9.2
04/19/23 Math 120 - KM 37
1
9
1
4
5 22
yx
13
1
2
52
2
2
2
yx
),( 45
),( 13 ),( 17
),( 25
9.2 Example: center not at the origin
),( 15
9.2
04/19/23 Math 120 - KM 38
324936 22 yx
1
63 2
2
2
2
yx
9.2 Example: Put in Standard Form First
1369
22
yx
324324
324
9
32436 22
yx
9.2
04/19/23 Math 120 - KM 39
),( 03),( 03
),( 60
),( 60
9.2 Example continued:Put in Standard Form First
),( 00
1
63 2
2
2
2
yx
9.2
04/19/23 Math 120 - KM 40
9.3 The Hyperbolait fits outside the box
9.3
04/19/23 Math 120 - KM 41
9.3 The HyperbolaSTANDARD FORM
12
2
2
2
b
ya
x
12
2
2
2
a
xb
y
9.3
04/19/23 Math 120 - KM 42
1.Fundamental Rectangle
2.Asymptotes
3.Vertices (if x2 – y2…)
4.Sketch
9.3 Hyperbola: x2 is first
9.3
04/19/23 Math 120 - KM 43
1416
22
yx
1
24 2
2
2
2
yx
),( 04),( 04
),( 20
),( 20
9.3 Example x2 is first
9.3
04/19/23 Math 120 - KM 44
1.Fundamental Rectangle
2.Asymptotes
3.Vertices (if y2 – x2…)
4.Sketch
9.3 Hyperbola: y2 is first
9.3
04/19/23 Math 120 - KM 45
116
x
4
y 22
),( 04),( 04
),( 20
),( 20
1
42 2
2
2
2
xy
9.3 Example y2 is first
9.3
04/19/23 Math 120 - KM 46
9.3 The HyperbolaNONSTANDARD FORM
numberpositivexy
numbernegativexy
9.3
04/19/23 Math 120 - KM 47
9.3 The HyperbolaNONSTANDARD FORM
Example 1
4xyx y
-4 -1
-2 -2
-1 -4
0 N
1 4
2 2
4 1
9.3
04/19/23 Math 120 - KM 48
9.3 The HyperbolaNONSTANDARD FORM
Example 2
4xyx y
-4 1
-2 2
-1 4
0 N
1 -4
2 -2
4 -1
9.3
04/19/23 Math 120 - KM 49
“Conic sections are among the oldest curves, and is an oldest math subject studied systematically and thoroughly. The conics seems to have been discovered by Menaechmus (a Greek, c.375-325 BC), tutor to Alexander the Great. They were conceived in an attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle. The conics were first defined as the intersection of: a right circular cone of varying vertex angle; a plane perpendicular to an element of the cone. (An element of a cone is any line that makes up the cone) Depending the angle is less than, equal to, or greater than 90 degrees, we get ellipse, parabola, or hyperbola respectively. Appollonius (estimated c. 262-190 BC) (known as The Great Geometer) consolidated and extended previous results of conics into a monograph Conic Sections, consisting of eight books with 487 propositions. Quote from Morris Kline: "As an achievement it [Appollonius' Conic Sections] is so monumental that it practically closed the subject to later thinkers, at least from the purely geometrical standpoint." Book VIII of Conic Sections is lost to us. Appollonius' Conic Sections and Euclid's Elements may represent the quintessence of Greek mathematics.
Appolloniuswas the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola. A brief explanation of the naming can be found in Howard Eves, An Introduction to the History of Math. 6th ed. page 172.
In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, espcially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic sections is a rich classic topic that has spurred many developments in the history of mathematics.”
From the website:http://xahlee.org/SpecialPlaneCurves_dir/ConicSections_dir/conicSections.html”
Conics...2300+ years old?
04/19/23 Math 120 - KM 50
9.4
9.4
04/19/23 Math 120 - KM 51
Think of the Possibilities!
9.4
04/19/23 Math 120 - KM 52
Where will they meet?
23
2
xy
xy
9.4
04/19/23 Math 120 - KM 53
Where will they meet - exactly?
23
2
xy
xy
232 xx0232 xx021 )x)(x(
1x or 2x ),(),,( 211 4
9.4
04/19/23 Math 120 - KM 54
Where will they meet - exactly?
5
252
22
xy
yx
2552 xx0202 xx054 )x)(x(
4x or 5x
),(),,(),,( 544 3 3 09.4
04/19/23 Math 120 - KM 55
Where will they meet - exactly?
9
922
22
yx
yx
182 2 x92 x
3x
),(),,( 33 0 0
9.4
04/19/23 Math 120 - KM 56
How about a really tough one?
4
204 22
xy
yx
204
42
2
x
x
02064
22
xx
02064 24 xx9.4
04/19/23 Math 120 - KM 57
How about a really tough one?
Continued...
02064 24 xx
4
204 22
xy
yx
06420 24 xx
0164 22 )x)(x(
42 x 162 xor9.4
04/19/23 Math 120 - KM 58
How about a really tough one?
Continued...
4
204 22
xy
yx
42 x 162 xor
2x 4xor
),(),,(),,(),,( 4422 2 2 1 1
9.4
04/19/23 Math 120 - KM 59
That’s All for Now!
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