maths course exercises

Liceo Scientifico Galileo Ferraris - Taranto

the hyperbola

in accordo con il

Ministero dell’Istruzione, Università, Ricerca e sulla base delle

Politiche Linguistiche della Commissione Europea

percorso formativo a carattere tematico-linguistico-didattico-metodologico

scuola secondaria di secondo grado

teacher

Rosanna Biffi

2

the hyperbola

Indice Modulo

Strategies - Before

• Prerequisites

• Linking to Previous Knowledge and Predicting con questionari basati su stimoli relativi alle conoscenze pregresse e alle ipotesi riguardanti i contenuti da affrontare

• Italian/English Glossary

Strategies – During

• Video con scheda grafica • Keywords riferite al video attraverso esercitazioni mirate • Conceptual Map

Strategies - After

• Esercizi: � Multiple Choice � Matching

� True or False � Cloze � Flow Chart

� Think and Discuss

• Summary per abstract e/o esercizi orali o scritti basati su un questionario e per esercizi quali traduzione e/o dettato

• Web References di approfondimento come input interattivi per test orali e scritti e per esercitazioni basate sul Problem Solving

Answer Sheets

3

the hyperbola

1

Strategies Before Prerequisites

The circle

Hyperbola

Cartesian plane The ellipse

Cartesian coordinates

Distance formula

Midpoint formula

Symmetric point accross the x–axis

Symmetric point accross the y-axis

Symmetric point accross the origin

Definitions and equation

Foci and axes

Horizontal and vertical ellipse

Eccentricity

Limit cases

Definition and equation Determination of centre and radius Position of a straight-line relative a circle Degenerate circle

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the hyperbola

2

Strategies Before

Linking to Previous Knowledge and Predicting

• Do you know the definition of conic section?

• Do you know the properties of the absolute value?

• Are you able to find the equaton of an ellipse centred at the origin, known

the foci and the major axis?

• Are you familiar with the concept of axial symmetry?

• Are you familiar with the concept of central symmetry?

• Do you know what is an asymptot?

• Do you know the Pythagorean Theorem?

• Do you know the definition of eccentricity of a circle and an ellipse?

• Are you familiar with the concept of degenerate conic?

• Are you able to solve a system of fourth degree with specific substitutions?

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the hyperbola

3

Strategies Before

Italian/English Glossary

Allineato Aligned

Antiorario Counterclockwise

Ascissa Abscissa

Asintoto Asymptote

Asse Axis (pl. axes)

Asse di simmetria Symmetry-axis

Asse focale Focal-axis

Asse non trasverso Conjugate-axis

Asse trasverso Transverse-axis

Attraversare To cross

Centro Centre

Coordinata Coordinate

Curva Curve

Denominatore Denominator

Diagonale Diagonal

Doppio cono (indefinito) Double cone (infinite)

Due metà Halves

Eccentricità Eccentricity

Ellisse Ellipse

Equilatera Equilateral

Estremi di un segmento Endpoints

Falda Fold

Forma canonica Canonical form

Formula Formula

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the hyperbola

Fuoco Focus (pl.foci)

Grafico Graph

Immaginario Imaginary

Infinito Endless

Intersezione con l’asse x (y) x (y)- intercept

Inversamente proporzionale Inversely proportional

Iperbole Hyperbola

Iperboloide Hyperboloid

Legge Law

Lunghezza Length

Nullo Null

Orario Clockwise

Ordinata Ordinate

Orizzontale Horizontal

Parabola Parabola

Perpendicolare Perpendicular

Punto medio Middle point

Rami Branches

Rapporto Ratio

Relazione Relation

Rettangolo Rectangle

Sostituzione Substitution

Tangente Tangent

Termine Term

Valore assoluto Absolute value

Vertice Vertex (pl. vertexes o vertices)

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the hyperbola

4

Strategies During

Keywords

1) Circle the conic sections:

straight-line – sphere – circle – pyramid – ellipse – circumference – hyperbola –

cylinder – cone - parabola

2) Completion:

• The hyperbola is a ………………………….

• It is characterized by two fixed points called…………... and two axes.

• The …………………… doesn’t contain the foci and its endpoints are the vertexes.

• The line passing the centre and perpendicular to the transverse axis is

the……………………….

• The …………………is obtained by the ratio of the……………………to the measure

of the transverse axis.

• The lines that contain the diagonals of the rectangle whose sides measure

………and…….., are the …………………………of the hyperbola.

If a=b, the measure of the conjugate and transverse axes is the same, then the

hyperbola is called ……………………….

________________________________________________________________

transverse-axis, eccentricity, foci, 2b, asymptotes, equilateral, conic section,

conjugate-axis, 2a, focal length

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the hyperbola

5

Strategies During

Conceptual Map

Complete the conceptual map using the following words and relations:

e > 1 b2x2-a2y2=a2b2

0<e<1

forming 2 branches

with both halves of

the cone

b2x2+a2y2=a2b2

a>b

b2x2-a2y2=-a2b2

b2x2+a2y2=a2b2

b>a

forming an angle

with the base

plane of the cone

x2+y2=r2

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the hyperbola

Conic

sections

perpendicular

to the cone

axis

obtained by slicing a right circular double cone with a plane

circle hyperbola ellipse parabola

centred at the origin

x-axis

foci

major axis 2a

minor axis 2b

eccentricity

e=0

transverse axis 2a

conjugate axis 2b

or

y-axis

major axis 2b

minor axis 2a

transverse axis 2b

conjugate axis 2a

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the hyperbola

6

Strategies After

Multiple Choice

1) The vertexes of the hyperbola 9y2 - 4x2= -36 are: a. (0,-2) (0,2) b. (-3,0) (3,0) c. (-2,0) (2,0) d. (0,-3) (0,3)

2) What is the equation of the focal axis of the hyperbola x2 - 16y2 = -144?

a. y=0 b. x=0

c. y= 13

d. y=2 13

3) The eccentricity of the hyperbola 4x2 - 25y2 = -100 is:

a. 0.93 b. 1.07 c. 2.7 d. 0.38

4) xy=-6 is the equation of a a. equilateral hyperbola b. horizontal hyperbola c. vertical hyperbola d. none of these 5) The length of the semi transverse axis of a hyperbola is 4, the eccentricity is 2.

What is the distance from a focus of the hyperbola to the vertex?

a. 4 b. 1.1 c. -4 d. none of these 6) What are the asymptotes of the hyperbola 36x2 - y2 = -144?

a. y=±8x

b. x=±8y

c. x=0 v y=0

d. y=±6x

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the hyperbola

7) What is the length of the semi conjugate axis of the hyperbola 35x2 - y2 =140?

a. 35 b. 2 c. 4

d. 352

8) What are the foci of the hyperbola x2 - 16y2 = -16?

a. ( 17± ,0) b. (0, ± 17) c. (0, 17± )

d. ( 17± ,0)

9) What are the y - intercepts of the hyperbola 25x2 - y2 = 25?

a. (-5,0) (5,0)

b. do not exist

c. (0,-5) (0,5)

d. (0,-25) (0,25)

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the hyperbola

7

Strategies After

Matching

Match the words on the left with the correct definition on the right:

1. Transverse axis

2. Hyperbola

3. Foci

4. Vertexes

5. Eccentricity

6. Asymptot

a. The geometric locus of points P which moves so that, the

difference between the

distances from P to two fixed

points, called foci, is a

constant

b. Endless tangent line to a curve

c. The intersection points

between the hyperbola and

the line passing through foci

d. Two fixed points on the

interior of a hyperbola used in

the formal definition of the

curve

e. The segment whose endpoints are the vertexes of a

hyperbola

f. The ratio of the focal length to the measure of the transverse

axis

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the hyperbola

8

Strategies After

True or False

State if the following sentences are true (T) or false (F).

1. The eccentricity of a hyperbola can be equal to 0.

2. The foci belong to the transverse axis.

3. The coordinates of the foci of a vertical hyperbola in canonical

form are ( 22 ba +± ,0).

4. The equation xy=-4 represents a hyperbola.

5. The equation 121

22

=−

+− k

y

k

x represents a hyperbola for

1<k<2.

6. Given a hyperbola in canonical form, if the foci are on the y-axis, then into the equation we find a < b.

7. The asymptotes of a hyperbola are tangents to the vertexes.

8. Two different hyperbolas have always different asymptotes.

9. Every hyperbola has always 2 symmetry axes.

10. The focal length of a hyperbola is the product between the eccentricity and the measure of the transverse axis.

� T � F � T � F

� T � F

� T � F � T � F � T � F � T � F � T � F � T � F � T � F

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the hyperbola

9

Strategies After

Cloze

Complete the text.

The hyperbola is an open curve with 2 ............ [1], the intersection of a plane,

with both........ [2] of a double-cone.

In a cartesian plane:

the hyperbola is the geometric locus of points......... [3] which moves so that, the

......... [4] between the ............ [5] from P to two fixed points, called foci, is a

constant.

If the hyperbola is horizontal, the x-intercept are called ........... [6].

The segment whose endpoints are the vertexes is called ..... ..... [7].

The distance between the two foci is called ..... ..... [8].

The midpoint of the transverse axis is the.......... [9] of the hyperbola.

The conjugate axis is the segment passing the centre and.......... [10] to the

transverse axis.

If the 2 foci are vertically aligned, the............axis [11] is on the x-axis.

The asymptotes of the hyperbola are endless.........lines [12] passing the origin.

The eccentricity of a hyperbola is the ratio of the......... length [13] to the

measure of the transverse axis.

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the hyperbola

10

Strategies After

Flow Chart

Complete the flow chart referring to the position of a straight-line in

relation to an ellipse. You can use the terms listed below: hyperbola,

ellipse, circle

false

start

e > 0

true

hyperbola,

eccentricity input

false

0 < e <1

true

output

end

output

16

the hyperbola

11

Strategies After

Think and Discuss

The following activity can be performed in a written or oral form. The teacher

will choose the modality, depending on the ability (writing or speaking) that

needs to be developed.

The contexts in which the task will be presented to the students are:

A) the student is writing an article about the ellipse

B) the student is preparing for an interview on a local TV about the ellipse

The student should:

1) Choose one of the following topics:

• The hyperbola eccentricity and the comparison with the one of the other conics

• The hyperbola in architecture • The Boyle’s law

2) Prepare an article or a debate, outlining the main points of the argument, on

the basis of what has been studied.

3) If the written activity is the modality chosen by the teacher, the student

should provide a written article, indicating the target of readers to whom the

article is addressed and the type of magazine / newspaper / school magazine

where the article would be published.

4) If the oral activity is the modality chosen by the teacher, the student should

present his point of view on the topics to the whole class and a debate could

start at the end of his presentation.

17

the hyperbola

12

Strategies After

Summary

A hyperbola is an open curve with two branches, the intersection of a plane with

both halves of a double cone. The hyperbola belongs to a family of curve

including parabolas, ellipses, circles.

The hyperbola is the geometric locus of points P which moves so that, the

difference between the distances from P to 2 fixed points, called foci, is a

constant, that is | PF1- PF2 | = 2a

The equation of the hyperbola centred at the origin is

where c is the abscissa of each focus and b2=c2–a2 , being c>a.

If the x-term is positive, it means that the hyperbola is horizontal or opening

East-West.

The x-intercepts of this curve are given by the points – a and a, that are called

vertexes of the hyperbola.

The transverse axis is the segment whose endpoints are the vertexes of the

hyperbola. Its measure is 2a.

The line passing the centre and perpendicular to the transverse axis is the

conjugate axis.

The centre of the hyperbola is the midpoint of the segment connecting the foci

or the vertexes.

It is a symmetry point for this curve. The coordinate axes are symmetry axes. The points of ordinates –b and b are imaginary y-intercepts.

In fact, the hyperbola doesn’t have inner points at the band delimited by the

vertical lines x = - a and x = a, then the curve is formed by 2 branches or

arms.

The positive number “b” is called measure of the conjugate semi-axis.

The horizontal lines passing the ordinates – b and b, with the vertical lines

passing the abscissas - a and a , form a rectangle whose sides measure 2a and

2b.

The lines that contain the diagonals of the rectangle are the asymptotes of the

hyperbola.

These asymptotes pass the origin and their equations are of type y = mx.

12

2

2

2

=−b

y

a

x

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the hyperbola

If the 2 foci are vertically aligned, the x-term is negative and the equation of the

hyperbola becomes as follows:

In this case the transverse axis is on the y-axis and its length is 2b.

It means that the hyperbola is opening North-South.

The equations of the asymptotes never change.

We define eccentricity of the hyperbola, the ratio of the focal length to the

measure of the transverse axis. This ratio is denoted by “e”, that is e = c/b.

In this case the number “e” is always greater than 1 and defines the hyperbola

opening.

The more the number “e” is over 1, that is the foci move away from the

vertexes, the more the hyperbola opens.

If a=b, the measure of the conjugate and transverse axes is the same, in this

case the curve is called equilateral hyperbola.

If we turn this curve 45° around the centre, counterclockwise or clockwise, the

asymptotes coincide with the coordinate axes and the equation becomes xy=k,

where k≠ 0. It is an equilateral hyperbola referred to the asymptotes.

We find the graph of this curve in the Boyle’s law: “ For a fixed mass of gas, at

constant temperature, the pressure and the volume are inversely proportional,

that is PV = k”.

Finally, we obtain conic sections by means of light; in fact, if we approach the

wall with a torch, we’ll see that the torch forms a cone of light that is projected

over the wall, which acts as a plane that crosses it.

By changing the slope of the torch, the projection of the light takes, in turn, the

form of a circle, an ellipse, a parabola or a hyperbola.

1) Answer the following questions. The questions could be answered

in a written or oral form, depending on the teacher’s objectives. a. How do you obtain a hyperbola?

b. What is the difference between an ellipse and a hyperbola?

c. What is the difference between a vertical and horizontal hyperbola?

d. Which symmetries does an hyperbola have?

e. What is the relationship between the coordinates of foci and the length of semi

axes?

12

2

2

2

−=−b

y

a

x

19

the hyperbola

f. What is the definition of eccentricity of a hyperbola?

g. Illustrate the procedure to obtain an equilateral hyperbola referred to the

asympotes.

h. Illustrate the “Boyle’s law”.

2) Write a short abstract of the summary (max 150 words) highlighting

the main points of the video.

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the hyperbola

Web References

An interactive math dictionary with many math words, math terms, math

formulas, pictures, diagrams, tables, and examples

http://mathworld.wolfram.com

http://www.mathwords.com

Encyclopedia of mathematics

http://www.britannica.com/EBchecked/topic/279494/hyperbola

http://en.wikipedia.org/wiki/Boyle%27s_law

Website designed to provide parents and classroom teachers with the means to

better employ visual imagery.

http://www.visualmathlearning.com

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the hyperbola

13

Activities Based on Problem Solving

a) Write the equation of the hyperbola, relative to the center and axes, passing

through the points A( 24 ,3) and B(- 34 ,3 2 ), with the foci on the x-axis. Verify, with the calculation, that a hyperbola with the foci on y-axis, passing

trough A and B, doesn’t exist .

b) Write the equation of the hyperbola, knowing that the foci are in the points

( 5± ,0) and the asymptotes are the lines y= x21± .

c) The vertexes of an hyperbola are ( 4± ,0), the foci are (± 41 ,0); find its equation.

d) Given the hyperbola x² - 3y² = 3, determine k so that the line y= kx+1 results tangent to the curve.

e) A ABCD square, with the sides parallel to the coordinate axes, has its vertexes on the hyperbola 9x2-y2=9. How does the area of the square measure?

f) From the point P(0,-1), conduct the tangent lines to the hyperbola 5x2-3y2=15.

g) Find the intersections of the hyperbola 16x2-25y2=400 with the circle centred at

the origin and radius equal to 108 .

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the hyperbola

Answer Sheets

Keywords:

1) circle – ellipse – hyperbola – parabola

2) conic section, foci, transverse axis, conjugate axis, eccentricity, focal length, 2a,

2b, asympotes, equilateral

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the hyperbola

Conceptual Map:

Conic

sections

perpendicular

to the cone

axis

forming an angle

with the base

plane of the cone

e > 1

obtained by slicing a right circular double cone with a plane

circle hyperbola ellipse parabola

centred at the origin

x-axis

b2x2-a2y2=a2b2

x2+y2=r2

foci

major axis 2a

minor axis 2b

eccentricity

0<e<1 e=0

forming 2 branches

with both halves of

the cone

transverse axis 2a

conjugate axis 2b

b2x2+a2y2=a2b2

a>b

or

y-axis

b2x2-a2y2=-a2b2

major axis 2b

minor axis 2a

transverse axis 2b

conjugate axis 2a

b2x2+a2y2=a2b2

b>a

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the hyperbola

Multiple Choice:

1b, 2b, 3c, 4a, 5a, 6d, 7d, 8c, 9b

Matching:

1e, 2a, 3d, 4c, 5f, 6b

True or False:

1F, 2F, 3F, 4T, 5T, 6F, 7F, 8F, 9T, 10T

Cloze:

[1] branches [2] halves [3] P [4] difference [5] distances [6] vertexes

[7] transverse axis [8] focal length [9] centre [10] perpendicular [11] conjugate

[12] tangent [13] focal

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the hyperbola

Flow Chart:

false

start

e>0 true

circle

hyperbola,

eccentricity input

false

0<e<1

ellipse

true

output

end

hyperbola

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the hyperbola

Activities Based on Problem Solving:

a. 9x2-16y2=144

b. x2-4y2=20

c. 25x2-16y2=400

d. k=32±

e. 29

f. y= 12 −± x

g. (± 20,4 15 ); (± 20,-4 15 )

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