(3) (4) The least positive integer solution of (3) is

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1 Meea K et al.; Sch. J. Phs. Math. Stat., 06; Vol-; Issue- (Feb); pp-5-9 Scholars Joural o Phsics, Mathematics ad Statistics Sch. J. Phs. Math. Stat. 06; ():5-9 Scholars Academic ad Scietiic Publishers (SAS Publishers) (A Iteratioal Publisher or Academic ad Scietiic Resources) ISSN 9-80 (Prit) ISSN (Olie) O The Biar Quadratic Diophatie Equatio =0 K. Meea, S. Vidhalakshmi, A. Nivetha Former VC, Bharathidasa Uiversit, Trich, Tamiladu, Idia Proessor, Dept. o Mathematics, SIGC, Trich, Tamiladu, Idia M. Phil Scholar, Dept. o Mathematics, SIGC, Trich, Tamiladu, Idia *Correspodi Author: S. Vidhalakshmi ts.maths.i@mail.com Abstract: The biar quadratic equatio represets a hperbola. I this paper we obtai a sequece o its iteral solutios ad preset a ew iteresti relatios amo them. Kewords: Biar quadratic equatio, iteral solutios. INTRODUCTION The biar quadratic Diophatie equatios (both homoeeous ad o-homoeeous) are rich i variet [,,, 4, 5, 6]. I the biar quadratic o-homoeeous equatios represeti hperbolas respectivel are studied or their o-zero iteral solutios. These results have motivated us to search or iiitel ma o-zero iteral solutios o aother iteresti biar quadratic equatio ive b The recurrece relatios satisied b the solutios ad are ive. Also a ew iteresti properties amo the solutios are ehibited [, 8, 9, 0,,,, 4, 5, 6]. METHOD OF ANALYSIS The Diophatie equatio represeti the biar quadratic equatio to be solved or its o-zero distict iteral solutio is () Note that () is satisied b the ollowi o-zero iteer pairs,, 4,,,, 4, 4 However, we have other solutios or (), which are illustrated below: Solvi () or, we have 4 () Let 4 Multipli the above equatio b o both sides ad perormi a ew calculatios, we have 49 () where (4) The least positive iteer solutio o () is 0, 0 4 Now, to id the other solutio o (),cosider the pellia equatio (5) whose udametal solutio is ~ ~ 0, 0, The other solutios o (5) ca be derived rom the relatios ~, ~ where Available Olie: 5

2 Meea K et al.; Sch. J. Phs. Math. Stat., 06; Vol-; Issue- (Feb); pp-5-9 Appli the lemma o Brahmaupta betwee relatio + 0, 0 &, ~, the other solutios o () ca be obtaied rom the Available Olie: 6 ~ (6) + () Taki positive si o the R.H.S o () ad usi (4),(6)&(), the o-zero distict iteer solutios o the hperbola () are obtaied as ollows (8),,,,5... (9) The recurrece relatios or, are respectivel A ew umerical eamples are ive i table below Table : Numerical Solutios Some relatios satisied b the solutios (8) & (9) are as ollows Each o the ollowi epressios is a ast umber i) ii)

3 Meea K et al.; Sch. J. Phs. Math. Stat., 06; Vol-; Issue- (Feb); pp-5-9 REMARKABLE OBSERVATIONS ) B cosideri suitable liear trasormatios betwee the solutios o (), oe ma et iteer solutios or hperbolas Eample ) Deie 4 6 8, Y 4 4 Note that the pair (, Y) satisies the hperbola Y Eample ) Deie , Y Note that the pair (, Y) satisies the hperbola Y 49 ) B cosideri suitable liear trasormatios betwee the solutios o (), oe ma et iteer solutios or parabolas Eample ) Deie 4 6 8, Y 4 4 Note that the pair (, Y) satisies the parabola Y Eample 4) Deie , Y Note that the pair (, Y) satisies the parabola Y Solvi () or, we have 8 49 (0) Let 8 49 Multipli the above equatio b o both sides ad perormi a ew calculatios, we have Y 49 () where Y 4 () The least positive iteer solutio o () is 0, Y 0 4 Now, to id the other solutio o (), cosider the pellia equatio Y () whose udametal solutio is ~ ~ 0, Y 0 =, The other solutios o () ca be derived rom the relatios Y ~ ~, where Appli the lemma o Brahmaupta betwee relatio + 0,Y 0 &, ~, the other solutios o () ca be obtaied rom the Y~ (4) Y + (5) Taki positive si o the R.H.S o (0) ad usi (),(4)&(5), the o-zero distict iteer solutios o the hperbola () are obtaied as ollows 4 (6) Available Olie:

4 Meea K et al.; Sch. J. Phs. Math. Stat., 06; Vol-; Issue- (Feb); pp-5-9 0,,4,6... The recurrece relatios or, are respectivel A ew umerical eamples are ive i table below, () Table : Numerical Solutios Some relatios satisied b the solutios (6) & () are as ollows Each o the ollowi epressios is a ast umber i) ii) REMARKABLE OBSERVATIONS B cosideri suitable liear trasormatios betwee the solutios o (), oe ma et iteer solutios or hperbolas Eample 5) Deie , Y 4 68 Note that the pair (, Y) satisies the hperbola Y Eample 6) Deie , Y Note that the pair (, Y) satisies the hperbola Y 49 B cosideri suitable liear trasormatios betwee the solutios o (), oe ma et iteer solutios or parabolas Eample ) Deie , Y 4 68 Note that the pair (, Y) satisies the parabola Y Available Olie: 8

5 Meea K et al.; Sch. J. Phs. Math. Stat., 06; Vol-; Issue- (Feb); pp-5-9 Eample 8) Deie , Y Note that the pair (, Y) satisies the parabola Y CONCLUSION I this paper, we have made a attempt to obtai a complete set o o-trivial distict solutios or the ohomoeeous biar quadratic equatio. To coclude, oe ma search or other choices o solutios to the cosidered biar equatio ad urther, quadratic equatios with multi-variables. ACKNOWLEDGEMENT The iacial support rom the UGC, New Delhi (F.MRP-5/4 (SERO/UGC) dated March 04) or a part o this work is rateull ackowleded. REFERENCES. Baumath TS; A Moder Itroductio To Aciet Idia Mathematics, Wile Easter Limited, Lodo, Carmichael RD; The Theor O Numbers Ad Diophatie Aalsis, Dover Publicatios, New York, Dickso LE; Histor O The Theor O Numbers, Vol. II, Chelsia Publicati Co, New York, Mordell LJ; Diophatie Equatios Acadamic Press, Lodo, Niel PS; The Alorithm Resolutios O Diophatie Equatios, Cambride Uiversit, Press, Lodo, Tela SG; Number Theor, Tata, Mc Graw-Hill Publishi Compa, New Delhi, Gopala MA, Parvath G; Iteral Poits O The Hperbola Atarctica J. Math, 00; (): Gopala MA; Vidhalakshmi S, Sumathi G, Lakshmi K; Iteral Poits O The Hperbola Bessel J. Math., 00; (): Gopala MA; Vidhalakshmi S; O The Diophatie Equatio Acta ciecia Idia, 00; IIIM(): Gopala MA, Gokila, Vidhalakshmi S, Devibala S; O The Diophatie Equatio 4,Acta Ciecia Idia, 00; IIIM(5): Gopala MA, Jaaki G; Observatio O. Impact J. sci. Tech., 008; (): Gopala MA, Shamuaadham P, Vijaashakar A; O Biar Quadratic Equatio Acta ciecia Idia, 008; IVM(4): Gopala MA, Vidhalakshmi S, Lakshmi K, Sumathi G; Observatio O Diophatus J. Maths., 0; (): Mollio RA; All Solutios O The Diophatie Equatios D Far East J Math. Sci. Speical., 998; III: Vidhalakshmi S, Gopala MA, Lakshmi K; O Biar Quadratic Equatio Scholar Joural O Phsics, Mathematics Ad Statistics, 04; (): Vidhalakshmi S, Gopala MA, Lakshmi K; Iteer Solutios O The Biar Quadratic Equatio 5 0. Iteratioal Joural O Iovative Sciece Eieeri & Techolo, 004; (6): Available Olie: 9