unit five conics math 621b 18 hours - gov.pe.ca · introductory conics lesson ... that describes...

114

UNIT FIVE

CONICS

MATH 621B

18 HOURS

Revised April 9, 02

115

Fun Activity

Take a page of paper and roll it into a tube. Put it to your eye and look through it but keep the other eyeto it. Focus on something in the distance. Put your other hand (palm facing you) touching the edge of thetube and close to its end. What do you see? Try making other shaped tubes, triangles, rectangles, etc. tosee if the same thing happens.

Explanation:Your eyes see things from slightly different views. Your brain mixes the two views into a composite full-dimensional view. With the paper telescope in place you are confusing your brain. One eye is sending anarrow view of the world to the brain while the other is sending a normal full-width view of an object(your hand) to the brain. Now the brain is confused and must make a very difficult compromise. It takesthe hole ( what ever the shape) from one eye and overlays it on the hand. The result; a hole in your hand.

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Introductory Conics Lesson

E40 demonstrate the locus of conic sections using concrete materials

ActivityIn a darkened room aim a flashlight at a wall so that it is perpendicular to the wall and about 15 cm fromthe wall. What three dimensional shape is the light radiated in?

What two dimensional shape is projected onto the wall? What happens to the shape as the flashlight ismoved directly closer to or farther away from the wall?

Now slowly move the flashlight in a circular path with respect to the wall. What two dimensional shapeis now observed on the wall?

Continue to turn the flashlight so that the angle that it makes with the wall continues to decrease. Whattwo dimensional shape is now observed on the wall?

Continue to turn the flashlight so that it is parallel and very close to the wall. What two dimensionalshape is now observed on the wall?

Read p.134 and write to explain how the diagrams on the bottom of p.134 relate to the above activity.

Research why these projections are called conic sections.

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SCO: By the end of grade12, students will beexpected to:

E18 determine the equation of a circle with arbitrary centre and radius

E19 recognize and use equations that describe circles and ellipses anywhere 2-D space

E22 use technology to graph circles with a given centre and radius

C23 express transformations algebraically and with mapping rules

Elaborations - Instructional Strategies/SuggestionsThe Circle (3.3)

Challenge student groups to do the Introductory Conics Lesson.

Note to Teachers: The diagrams on p.134 model the IntroductoryLesson. The shapes are generated in the order on the bottom of p.134from (a) 6 (b) 6 (d) 6 (c).

Invite student groups to read and discuss p.138-140 and make notes intheir journals on:

< the definition of a circle < the standard form of the equation of a circle < the general form of the equation of a circle x2 + y2 + Dx + Ey + F = 0 < completing the square to get the standard form

Students will have seen Pythagoras Theorem in different forms. Inearlier years they would have used c2 = a2 + b2 . In Math 521B students

encountered the theorem in the form;

d x x y y= − + −( ) ( )2 12

2 12

This form is used to calculate the length of a line segment on a Cartesiancoordinate plane. In this section we will modify this second form by

replacing d with r, (x2,,y2) with (x,y) and (x1,y1) with (h,k). As wellboth sides of this formula will be squared, yielding the standard form:

( ) ( )x h y k r− + − =2 2 2

where (h,k) is the centre of the circle r is the radius of the circle

Students should become familiar with drawing circles on the TI-83using the equation editor and the draw circle feature (see Ex. 4 p.140).

Examples of conics:roof of Paris subway, Royal Albert Hall, St. Paul’s Cathedral, room inU.S. Capitol, headlight mirrors, satellite dishes, elliptical galaxies, orbitsof planets, comets, electrons, lithotripter.

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Worthwhile Tasks for Instruction and/or Instruction Suggested Resources

The Circle (3.3)

Research/PresentationUse the internet to find examples in nature and man-madeobjects of the conics sections.

Pencil/PaperDetermine the equation in standard form for the circle withcentre (!3,2) and passing through (4,!3). Expand to find thegeneral form.

Pencil/PaperFind the centre and radius for the circle defined by: x2 + y2 + 6x !10y !2 = 0.

PresentationCreate an art design of circles using the Circle Draw featureof the TI-83.

DiscussionWhat is the equation for each of the translations of the circle:x2 + y2 = 49. Sketch each of the graphs.a) (x,y) 6 (x + 3, y ! 2)b) (x,y) 6 (x ! 5, y ! 1)c) (x,y) ! (x ! 1, y + 3)

ApplicationPortable autonomous digital seismographs (PADS) are used tomeasure the aftershocks of earthquakes. A PADS device located 10 km east and 6 km south of Vancouver records theseismic activity of a quake 15 km away. Write an equationthat describes all the possible locations of this earthquake’sepicentre.

The Circle (3.3)

Warm Up p.133 #17-28Mental Math p.133 #35-40

Math Power 12 p.141 #1-35 odd, 36,39,41,43,45,50-54 56,58

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E19 recognize and use equations that describe circles and ellipses anywhere 2-D space

E35 translate between graphical and symbolic representation of circles, ellipses, parabolas and hyperbolas


Elaborations - Instructional Strategies/SuggestionsThe Ellipse (3.4)

Challenge student groups to do the “Ellipse Discovery Experiment” atthe end of the unit.

Invite student groups to read and discuss p.143-149. The general formfor the ellipse:

Ax2 + Bxy + Cy2 +Dx + Ey + F = 0

where A … C A & C have the same sign A > C then the major axis is vertical A < C then the major axis is horizontal

Standard forms of ellipses are:

( ) ( )

( ) ( )

x ha

y kb

horizontal major axis

y ka

x hb

vertical major axis

where a b c

−+

−=

−+

−=

= +

2

2

2

2

2

2

2

2

2 2 2

1

1

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The Ellipse (3.4)

Research/ProjectDesign an elliptically shaped transom to go over an exteriordoor with glass sidelights. Take into account the total widthof the door system and the limiting factor of the ceilingheight.

ApplicationTo eliminate kidney stones, doctors use a tool called a“lithotripter” ( meaning stone crusher). It uses ultra-highfrequency shock waves moving through water to break up thestone. Once a patient has been x-rayed to accurately locatethe kidney stone, a person is immersed in an elliptically-shaped tank with the shock-wave emitter located at one focalpoint and the kidney stone at the other focal point. Theelliptical tank has a semi-major axis of 36 cm and a semi-minor axis of 20 cm. How far from the emitter should thekidney stone be?

The Ellipse (3.4)

Math Power 12 p.150 #1-10, 11-19 odd, 21-28Note: Do only 2 of 21-24 in generalform.

Applications p.152 # 39,40,42

Do ellipse worksheet at end of unit.

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E38 determine the equation of a hyperbola

Elaborations - Instructional Strategies/SuggestionsThe Hyperbola (3.5)

Student groups should do the Explore & Inquire on p.154. Invitestudent groups to read and discuss p.155-158.

Terms to become familiar with are:

< transverse axis < conjugate axis < asymptotes < vertices of a hyperbola

The general form of the equation of a hyperbola is:

Ax2 + Cy2 + Dx + Ey + F = 0 where A and C have opposite signs.

The standard form of the equation of a hyperbola is:

( ) ( )

( ) ( )

x ha

y kb

horizontal transverse axis

y ka

x hb

vertical transverse axis

−−

−=

−−

−=

2

2

2

2

2

2

2

2

1

1

The LORAN navigation system uses hyperbola theory to locate ships atsea. Today it is being phased out and the GPS system is gaining favour. See the “Applications of Hyperbolas Sheet” at the end of the unit to seethe theory behind these types of problems.

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The Hyperbola (3.5)

Pencil/Paper

For the hyperbola ( ) ( )x y+

−−

=2

163

91

2 2

determine the:a) coordinates of the centreb) coordinates of the verticesc) equations of the asymptotesd) sketch the asymptotes and the hyperbola

Group ActivityWrite the equation of the hyperbola in standard and generalform with centre (1,2), one focus at (1,5) and difference offocal radii 4.

PresentationUse the internet to find examples of hyperbola occurrence inthe real world.

The Hyperbola (3.5)

Math Power 12 p.159 # 1-13 odd, 14-18, 19-23 odd, 25-28,33,35, 41(a),(b), 52

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Elaborations - Instructional Strategies/SuggestionsThe Parabola (3.6)

Invite student groups to read and discuss p.165-167.

A parabola is the locus of all points in a plane that are equidistant froma line (the directrix) and a point not on that line ( the focus).

The general form of a parabola is:

Ax2 + Cy2 + Dx +Ey + F = 0 where either A or C = 0.

The standard form of a parabola is:

opening up or down: ( ) ( )x h p y k− = −2 4

opening left or right: ( ) ( )y k p x h− = −2 4

Key terms for students to understand are:

< axis of symmetry < vertex < focus < latus rectum

Note to Teachers: the expression “4p” , is the latus rectum. Look at thediagram for the parabola in the “Conic Theory” sheets at the end of theunit. The latus rectum is the width, if you like, of the parabola asmeasured along a line parallel to the directrix and passing through thefocus. It is a tool that will aid students in sketching a parabola withouthaving to get a detailed table of values.

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The Parabola (3.6)

Pencil/PaperSketch the parabola with focus (!2,2) and directrix y = !4. Write the equation for this conic in standard form.

Group ActivityFor the parabola, y2 !2x !4y + 14 = 0 determine thecoordinates of the focus and the vertex and the equation of thedirectrix. Sketch the graph of this conic.

DiscussionClassify each of the following as a specific conic:a) x2 ! 4x + 2y ! 6 = 0b) x2 + 2x + y2 ! 6y + 4 = 0c) 2x2 + 8x + 3y2 + 6y ! 29 = 0d) 3x2 + 4x ! 6y2 ! 8y ! 10 = 0

ActivityWrite x2 + 6x ! 4y ! 8y ! 11 = 0 in standard form. Thensketch a graph of this conic section.

The Parabola (3.6)

Math Power 12 p.167 # 1-13 odd, 15-31, 33-43 odd, 46,48

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Ellipse Discovery Experiment (3.4)

OBJECT: To construct and discover the definition of a closed curve.

APPARATUS: Two push pins, cardboard, coordinate grid, string, pencil, 30 cm ruler.

PROCEDURE: Make slip knots at each end of the string. Put the push pins inside eachloop and push the pins into the grid paper along either a horizontal orvertical grid line. Label the two points where you placed the push pins asF1 and F2. Place a pencil so that the string is stretched tight and draw thelocus of the resulting curve. Speculate on the name of this curve. Pick four points at random on the curve and measure the distance fromeach point to F1 and F2. record this data in the table below.

Distance to F1 Distance to F2 Sum of Distances

1st Point

2nd Point

3rd Point

4th Point

Use inductive reasoning to describe the pattern observed in the table above. Compare the shapeof your group’s figure with other groups. How are they the same? How are they different?

Research various sources (internet, texts, etc.) as to where these curves occur naturally or inman-made objects.

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Ellipse Worksheet (3.4)

In preparation for PEI hosting the Canada Summer Games in 2009 you are asked to design a newstadium. The stadium is to be formed by two concentric(same centre) ellipses. The outer ellipseis to be 260 m long and 220 m wide while the inner ellipse is to be 220 m long and 110 m wide.

a) Set up a coordinate system to illustrate this design, using the origin as the centre of the twoellipses. What are the equations for the ellipses?

b) The hosts wish to place a rectangular playing field that is 130 m long and 58 m wide in thecentre of the inner ellipse. Is this possible? Explain and show a diagram.

c) The area of an ellipse is Bab, where a and b are the horizontal and vertical semi-axes. To thenearest square metre, what area will the stands cover?

d) If, on average, each seat occupies 0.75 m2, how many seats are in the stadium? Theorganizers want to charge $12 per ticket for the opening ceremonies. If the opening ceremoniessell out, what is the total income from admissions?

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Applications of Hyperbolas (3.5)

Two rescue teams are trying to locate a lost child in a forest. The rescuers can hear the child’sshouts for help. If the teams are 9 km apart and Team 2 hears the shouts of the child 5 secondsafter Team 1. Use a coordinate system that has its origin at Team 1. Write an equationdescribing the possible locations of the child. Assume that the speed of sound is 343 m/s or .343 km/s.

Solution:

Using the formula d = rate × time we get the times for the sound to travel to the teams is d1 / aand d2 / a . These are 5 seconds apart and thus we get :

d d

d d km

2 1

2 1

343 3435

17. .

.

− =

− =

The above formula tells us that the distance between the two teams(the foci) is 1.7 km and if weassume it to remain the same, this is the definition for a hyperbola..

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Using the distance formula and the figure below:

d d

x y x y

x y x y

x y x y x y

x x y x y x y

x x y

x x y

x x x y

x x y

2 1

2 2 2 2

2 2 2 2

2 2 2 2 2 2

2 2 2 2 2 2

2 2

2 2 2 2

2 2 2

2 2

17

9 17

9 17

9 2 9 3 4

18 2 9 3 4

781 18 3 4

781 18 3 4

6100 2811 324 116

312 4 2811 116 6100

− =

− + − + =

− + = + +

− + = + + + +

− + = + + + +

− = +

− = +

− + = +

− + + =

.

( ) .

( ) .

( ) . .

. .

. .

( . ) ( . )

. ( )

. .

This is the equation that describes the locus of points where the child could be located. Noticethat it is the equation of a hyperbola.

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Examples of Conic Sections

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Conic Theory

A conic section is the locus (set of points) P determined by a line d (the directrix) and a point F ( thefocus) not on line d such that PF/Pd = e where “e” is non-negative.

The vertex is the point on the locus where PF and Pd are the shortest lengths.The axis of symmetry is the line through F perpendicular to “d”.The latus rectum is the distance between those two points on the conic which are also on the line throughF parallel to “d”.

Ellipse: 0 < e < 1

For ellipses PF < Pd

Hyperbola: e > 1

For hyperbolas PF > Pd

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Parabolas: e = 1

For parabolas PF = Pd

Notice the relationship between the lengths of PF and Pd in these three conics.You may have noticed that the circle has not been discussed. For a circle “e” = 0. What this means isthat PF/Pd = 0, which can be interpreted as having a directrix line at infinity. If you are interested infinding out more, do some reading on “ geometry in the projective plane”.

unit five conics math 621b 18 hours - gov.pe.ca · introductory conics lesson ... that describes...

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