Liceo Scientifico “G.Ferraris”Taranto

School Year 2011-2012

Maths course The hyperbola

UTILIZZARE SPAZIO PER INSERIRE FOTO/IMMAGINE DI RIFERIMENTO LEZIONE

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 curves including parabolas, ellipses, circles.

Conic section

| PF1- PF2 | = const

The hyperbola is 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.

Hyperbola as geometric locus

The equation of the hyperbola can be found by using

the distance formula:

Finding the hyperbola equation

given F1(-c ; 0) and F2(c ; 0) c > 0,

let P (x,y) such that | PF1- PF2 | = 2a a > 0

(c2-a2)x2-a2y2=a2(c2-a2) c > a

From this relation, after eliminating radicals and simplifying,

we obtain the hyperbola equation centred at the origin:

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

horizontal or opening East-West

If we place b2=c2–a2 into the previous equation,

we’ll obtain the following:

Hyperbola in canonical form

The hyperbola foci and vertexes

-a a

b

c

c

-c

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

that are called vertexes of the hyperbola.

The points of ordinates –b and b are imaginary y-intercepts.

-bc2=a2+b2

Pythaghorean Theorem

The hyperbola axes

- a ac-c

transverse axis

The transverse axis is the segment whose endpoints are the

vertexes of the hyperbola. Its measure is 2a.

The line passing the origin and perpendicular to the transverse

axis is the conjugate axis.

The hyperbola simmetries

It is a symmetry point for this curve. The coordinate axes are symmetry axes.

The centre of the hyperbola is

the midpoint of the transverse

axis that is the origin.

It ‘s also the midpoint of the

segment connecting the foci.

The positive number “b” is called measure of the conjugate

semi-axis.

Hyperbola position

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, as shown in the picture.

branche branche

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 diagonal of this rectangle has the same measure of the

focal length, 2c .

The lines that contain the

diagonals of the rectangle

are the asymptotes of

the hyperbola, they are

endless tangent lines.

These asymptotes pass the origin and their equations are of type

y = mx where m = b/a v m = - b/a.

The asymptotes of a hyperbola

a = 4 is the semi-transverse axis

b = 3 is the semi-conjugate axis

c = 5 is the distance from the centre to each focus

y = x are the asymptotes

Example

If a=4 , b=3 and the foci

are horizontally aligned,

the equation is:

43

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

the equation of the hyperbola becomes:

Vertical hyperbola

12

2

2

2

b

y

a

x

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 = 2c/2b, e = c/b.

Hyperbola eccentricity

This number “e” is always

greater than 1 and defines

the hyperbola opening.

e1 < e2

e1

e2

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

from the vertexes, the more the hyperbola opens.

Eccentricity variation

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

the same, then the hyperbola is called equilateral.

Turning this curve 45° around the centre, the asymptotes

coincide with the coordinate axes.

Equilateral hyperbola

equilateral hyperbola

referred to the asymptotesK>0

K<0

xy = k k≠0

Boyle’s law:

PV=k

The case of k>0 represents the law of the inverse proportionality.

Tuscany

Cooling towers of the geysers

KÕbe Port Tower-Japanby Vladimir Shukhov

Cathedral of Brasiliaby Oscar Niemeyer

Course TeacherRosanna Biffi

Linguistic Support Flaviana Ciocia

Performed byTeacher: Rosanna Biffi

Students: Arnesano Alessandro, Basile Giulia, Biondolillo Alessia, Bruno Marianna, D’andria Roberta, Manco Marcello

(Grade 5 D - Secondary High School)

Acknowledgement Marco Dal Bosco 

Headmaster

Technical Supporteni

Director Rosanna Biffi

 

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