1. Liceo Scientifico “G.Ferraris” Taranto School Year 2011-2012 Maths course The hyperbola UTILIZZARE SPAZIO PER INSERIRE FOTO/IMMAGINE DI RIFERIMENTO LEZIONE

2. 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

3. | PF 1 - PF 2 | = 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

4. The equation of the hyperbola can be found by using the distance formula: Finding the hyperbola equation given F 1 (-c ; 0) and F 2 (c ; 0) c > 0, let P (x,y) such that | PF 1 - PF 2 | = 2a a > 0 (c 2 -a 2 )x 2 -a 2 y 2 =a 2 (c 2 -a 2 ) c > a From this relation, after eliminating radicals and simplifying, we obtain the hyperbola equation centred at the origin:

5. If the x-term is positive, it means that the hyperbola is horizontal or opening East-West If we place b 2 =c 2 –a 2 into the previous equation, we’ll obtain the following: Hyperbola in canonical form

6. The hyperbola foci and vertexes -aa 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. -b c 2 =a 2 +b 2 Pythaghorean Theorem

7. The hyperbola axes - a a c-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.

8. 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.

9. 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

10. 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.

11. 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

12. 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:

13. If the 2 foci are vertically aligned, the x-term is negative and the equation of the hyperbola becomes: Vertical hyperbola 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.

14. 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. e 1 < e 2 e1e1 e2e2

15. The more the number “e” is over 1, that is the foci move away from the vertexes, the more the hyperbola opens. Eccentricity variation

16. 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 asymptotes K>0 K<0 xy = k k≠0

17. Boyle’s law: PV=k The case of k>0 represents the law of the inverse proportionality.

18. Tuscany Cooling towers of the geysers

19. K Õ be Port Tower-Japan by Vladimir Shukhov Cathedral of Brasilia by Oscar Niemeyer

20. Course Teacher Rosanna Biffi Linguistic Support Flaviana Ciocia Performed by Teacher: Rosanna Biffi Students: Arnesano Alessandro, Basile Giulia, Biondolillo Alessia, Bruno Marianna, D’andria Roberta, Manco Marcello (Grade 5 D - Secondary High School)

21. Acknowledgement Marco Dal Bosco Headmaster Technical Support eni Director Rosanna Biffi Copyright 2012 © eni S.p.A.