GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES
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1 GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES A STUDY ON THE HYPERBOLA x 9 0 S. Vidhyalakshmi, M.A. Gopala & T. Mahalakshmi*, Professor, Departmet of Mathematics, Shrimati Idira Gadhi College, Trichy-60 00, Tamil Nadu, Idia * Assistat Professor, Departmet of Mathematics, Shrimati Idira Gadhi College, Trichy-60 00, Tamil Nadu, Idia ABSTRACT This paper aims at obtaiig o-zero distict iteger solutios to the hyperbola represeted by the biary quadratic equatio x 9 0. A few iterestig relatios amog the solutios are preseted. Employig the solutios of the give hyperbola, solutios for other choices of hyperbolas ad parabolas are preseted. Keywords: Biary quadratic, o-homogeeous quadratic, hyperbola, Iteger solutios. 00 Mathematics Subject Classificatio: D09. I. INTRODUCTION The biary quadratic Diophatie equatios (both homogeeous ad o homogeeous) are rich i variety [-6]. I [7-8], the biary quadratic o-homogeeous equatios represetig hyperbolas respectively are studied for their o-zero itegral solutios. These results have motivated us to search for ifiitely may o-zero itegral solutios of yet aother iterestig biary quadratic equatio give by x 9 0.The recurrece relatios satisfied by the solutios x ad y are give. Also a few iterestig properties amog the solutios are exhibited. II. METHOD OF ANALYSIS The Diophatie equatio uder cosideratio is x 9 0 () Itroductio of the trasformatios x X, y Y () i () gives X Y 7YX 4 0 () Substitutio of the trasformatios, X u v, Y u v, u v 0 (4) i () leads to v u 4 (5) whose smallest positive iteger solutio is u 0, v 0 Now, cosider the pellia equatio v u (6) whose geeral solutio u ~, v~ ) is give by v ~ f, u ~ ( g 6 (C)Global Joural Of Egieerig Sciece Ad Researches
2 where g [( ) ( ) ] f [( ) ( ) ], 0,,,... Applyig Brahmagupta lemma betwee the solutios u, ) ad u ~, v~ ), we have ( v 0 0 ( f g u f g v Takig the advatage of () ad (4), the sequece of itegral solutios of () ca be writte as x 6 f 8 g (7) y ( f 5 g ) (8) Thus (7) ad (8) represet the o-zero distict itegral solutios of (). The above values of respectively the followig recurrece relatios. x 98x x 96 y 98y y 96, 0,,,.... Some umerical examples of x ad y satisfyig () are give i the Table : below. x ad y satisfy Table : Examples x y From the above table, we observe some iterestig relatios amog the solutios which are preseted below.. x values are positive ad y values are egative. Both values are odd.. Each of the followig expressios is a Nasty umber : ( 4x x 77) 45 ( 607x x4 0650) 5840 (5x 8y 9) ( 5585x 8y 807) 6 ( 7495x 8y ) Each of the followig expressios is a Cubical iteger: ( 4x x4 70x x 49) 70 ( 607x x x x ) (5x 8y 5x 4y 5) 6 (C)Global Joural Of Egieerig Sciece Ad Researches
3 ( 5585x 8y4 6755x 4y 7) ( 7495x 8y x 4y 8997) Each of the followig expressios is a Bi-Quadratic iteger: ( 4x 44 x45 496x 4x 7785) 70 ( 607x 44 x x 4x4 040) (5x 44 8y44 0x 5) ( 5585x 44 8y45 40x 597) ( 7495x 44 8y x ) Each of the followig expressios is a Quitic iteger: ( 4x 55 x56 670x 5x4 40x 0x 978) x55 x x 5x5 6070x x (5x 55 8y55 5x 40y 50x 80y 08) ( 5585x 55 8y56 795x 40y x 80y 89488) 7495x55 8y x 40y x 80y Relatios amog the solutios: x 6874x x x x y 8000x x y 60x 58658y x x 40840y y 080x 6y Remarkable Observatios (i) Employig liear combiatios amog the solutios of (), oe may geerate iteger solutios for other choices of hyperbolas which are preseted i the Table: below. Table : Hyperbolas S.NO Hyperbolas (X,Y) 08X Y x x, 7797x X Y x x 6070, 0049x (C)Global Joural Of Egieerig Sciece Ad Researches
4 X Y 08 5x 8y, 6y 9 4 X Y X Y x 8y 549x y 7495x 678x 8y 6y 559, , 6789 (ii) Employig liear combiatio amog the solutios of (), oe may geerate iteger solutios for other choices of parabolas which are preseted i the Table: below. Table: Parabolas S.NO Parabolas (X,Y) 5660X Y x x 77, 7797x X Y x x4 0650, 0049x X Y 08 5x 8y 9, 6y X Y x 8y 807, 549x y X Y x 8y , 678x 6y 6789 III. CONCLUSION I coclusio, oe may search for other patters of solutios ad their correspodig properties. REFERENCES. R.D. Carmichael, The Theory of Numbers ad Diophatie Aalysis, Dover Publicatios, New York (950).. L.E. Dickso, History of theory of umbers, vol., Chelsea publishig compay, New York (95).. L.J. Mordell, Diophatie equatios, Academic Press, Lodo (969). 4. S.G. Telag, Number Theory, Tata Mc Graw Hill Publishig Compay, New Delhi (996). 5. Nigel.P.Smart., The Algorithm Resolutios of Diophatie Equatios, Cambridge Uiversity Press, Lodo (999). 6. T.S.Baumathy, A Moder Itroductio to Aciet Idia Mathematics, Wiley Easter Limited, Lodo (995). 7. K. Meea, S. Vidhyalakshmi ad A. Nivetha O The Biary Quadratic Diophatie Equatio x 4xy y 4x 0. Scholars Joural of Physics, Mathematics ad Statistics,(),06, M.A. Gopala, V.Geetha ad D. Priyaka O The Biary Quadratic Diophatie Equatio x xy y 6x 0. Iteratioal Joural of Scietific Egieerig ad Applied Sciece,(4),Jue 05, (C)Global Joural Of Egieerig Sciece Ad Researches
5 9. M.A. Gopala, S. Vidhyalakshmi, J. Shathi ad S. Sugua O The Biary Quadratic Diophatie Equatio x 4xy y x 0. Bulleti of Mathematics ad Statistics Research,(),05, S. Vidhyalakshmi, M.A. Gopala ad S. Nadhii O The Biary Quadratic Diophatie Equatio x xy y 8x 0. Iteratioal Research Joural of Egieerig ad Techology, (4),July 05, S. Vidhyalakshmi, M.A. Gopala, A. Kavitha ad D.Mary Madoa O The Biary Quadratic Diophatie Equatio x xy y x 0. Iteratioal Joural of Scietific Egieerig ad Applied Sciece, (4),, Jue 05, -5.. S. Vidhyalakshmi, M.A. Gopala, R.Presea ad N.Christy O The Biary Quadratic Diophatie Equatio x xy y x 0. Iteratioal Joural of Applied Research, (8), 05, M.A. Gopala, S.Vidhyalakshmi, R. Presea ad K. Lakshmi O The Biary Quadratic Diophatie Equatio x xy y 0x 0. Iteratioal Joural of Egieerig, sciece ad Mathematics, 4(), Jue 05, K. Meea, S. Vidhyalakshmi, M.A. Gopala ad K. Aitha Iteger Poits O The Hyperbola x 5xy y 0x 0, Archimedes J. Math., 4(),04, M.A. Gopala, S. Vidhyalakshmi ad J. Shathi Itegral Poits o The Hyperbola x 4xy y x 0, Bulleti of Mathematics ad Statistics Research.,(),04, S. Vidhyalakshmi, A. Kavitha ad M.A. Gopala, Itegral Poits o The Hyperbola x 4xy y 5x 0. Iteratioal Joural of Iovative Sciece, Egieerig ad Techology,(7),Sep-04, S. Vidhyalakshmi, M.A. Gopala ad K.Lakshmi, Observatio o The Biary Quadratic Equatio x 8xy x y 6 0. Scholars Joural of Physics, Mathematics ad Statistics, (),04, S. Vidhyalakshmi, M.A.Gopala ad K.Lakshmi Iteger solutios of The Biary Quadratic Equatio x 5xy y x 0, Iteratioal Joural of Iovative Sciece, Egieerig ad Techology, (6), Aug-04, (C)Global Joural Of Egieerig Sciece Ad Researches
International Journal of Multidisciplinary Research and Development. M.A. Gopalan, A. Kavitha, G. Thamaraiselvi
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