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03/22/22 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

9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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Page 1: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 2: 9/22/2015Math 120 - KM1 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

Page 3: 9/22/2015Math 120 - KM1 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 3

Sections of a Cone ... continued

Page 4: 9/22/2015Math 120 - KM1 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 4

Degenerate Conic Sections

Page 5: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 5

9.1

Page 6: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 6

The Parabola

9.1

Page 7: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 7

A Parabolic ReflectorFor a Microphone

Can You Hear a Pin Drop?

9.1

Page 8: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 8

A Parabolic Archway

Architectural Parabola

9.1

Page 9: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 9

A Parabolic Headlight

Shine Your Light Forward

9.1

Page 10: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 10

Parabolic Shadows

9.1

Page 11: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 12: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 13: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 14: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 15: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 16: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 17: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 18: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 19: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 20: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 20

111 ,yxP

222 ,yxP

c

a

b

2212

21 yyxxd

22 bac

The Distance Formula

9.1

Page 21: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 22: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 22

9.1 MIDPOINT

9.1

Page 23: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 24: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 25: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 18:05 krm 11.2 25

With a COMPASS

How do I make a

circle ?

9.1 Circles

9.1

Page 26: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 27: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 27

9.1 x2 + y2 = 1

9.1

Page 28: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 28

9.1 (x + 2)2 + (y – 4)2 = 32

9.1

Page 29: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 29

9.1 x2 + (y + 4)2 = 25

9.1

Page 30: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 31: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 32: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 33: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 33

9.2 The Ellipse

9.2

Page 34: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 35: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 36: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 37: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 38: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 39: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 40: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 40

9.3 The Hyperbolait fits outside the box

9.3

Page 41: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 42: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 43: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 44: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 45: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 46: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 46

9.3 The HyperbolaNONSTANDARD FORM

numberpositivexy

numbernegativexy

9.3

Page 47: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 48: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 49: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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?

Page 50: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 50

9.4

9.4

Page 51: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 51

Think of the Possibilities!

9.4

Page 52: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 52

Where will they meet?

23

2

xy

xy

9.4

Page 53: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 54: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 55: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 56: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 57: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 58: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

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

Page 59: 9/22/2015Math 120 - KM1 Chapter 9: Conic Sections 9.1 Parabola (Distance Formula) (Midpoint Formula) Circle 9.2 Ellipse 9.3 Hyperbola 9.4 Nonlinear Systems

04/19/23 Math 120 - KM 59

That’s All for Now!