[Pandichelvi*, 4.(7): July, 2015] ISSN: (I2OR), Publication Impact Factor: 3.785

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1 [Pandichelvi*, 4.(7): Jul, 5] ISSN: (IOR), Publication Impact Factor: IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY INTEGRAL SOLUTIONS OF THE BINARY QUADRATIC EQUATION 4 7 V. Pandichelvi*, P. Sivakamaundari, M.A. Gopalan * Aitant Profeor, Department of Mathematic, UDC, Trich. Guet Lecturer, Department of Mathematic, BDUCC,Laludi, Trich. Profeor, Department of Mathematic, SIGC, Trich-. ABSTRACT The hperbola iven b 4 7 i analzed for it non-zero ditinct inteer olution. A few interetin relation amon the olution are preented. Emploin the olution of the equation under conideration,the inteer olution for a few choice of hperbola and parabola are obtained. KEYWORDS: Binar Quadratic, Hperbola, Parabola, Interal Solution, Pell Equation Mathematic ubject claification: D9. INTRODUCTION Diophantine equation of the form D where D i a iven poitive quare free inteer i known a pell equation and i one of the oldet Diophantine equation that ha intereted mathematician all over the world, ince antiquit, J.L. Larane proved that the poitive pell equation D ha infinitel man ditinct inteer olution where a the neative pell equation D doe not alwa have a olution. In [], an elementar proof of a ceriterium for the olubilit of the pell equation D where, D i an poitive non-quare inteer, ha been preented. For eample, the equation 3, 7 4 have no inteer olution, where a 5, have inteer olution. In thi contet, one ma refer [-]. In thi communication, the hperbola repreented b 4 7 i analzed for it nonzero ditinct inteer olution. A few interetin relation amon the olution are preented.emploin the olution of the equation under conideration,the inteer olution for a few choice of hperbola and parabola are obtained. METHODS FOR ANALYSIS The binar quadratic equation repreentin hperbola i 4 7 () We preent below two different pattern of non-zero ditinct inteer olution of (). PATTERN: Treatin () a a quadratic in and olvin for,we have 5 http: // 3 () International Journal of Enineerin Science & Reearch Technolo [89]

2 [Pandichelvi*, 4.(7): Jul, 5] ISSN: (IOR), Publication Impact Factor: Let 3 Y (3) where Y 3 (3a) The initial olution of (3) i, Y To find the other olution of (3),conider the pellian equation whoe eneral olution i iven b ~ where ~ f, Y Y (4) f (5 ) (5 ) (5 ) (5 ),,,,3,... Applin Brahmaupta lemma between, ) and ~ ~, Y ) the other inteer olution of (3) are iven b Y f f Y 3 ( Y ( Subtitutin the above value of, in(3a),() and takin the poitive in on the RHS of(),we et 5 f f which repreent the inteer olution of(). The recurrence relation atified b and are iven b 3 3, 4, 3, 5, 47 5 Some numerical eample of and atifin () are iven in the followin table. http: // Table : Numerical Eample International Journal of Enineerin Science & Reearch Technolo [9]

3 [Pandichelvi*, 4.(7): Jul, 5] ISSN: (IOR), Publication Impact Factor: A few interetin propertie between the olution are iven below:. ( 4 8) i a nat number ( i a perfect 4 +3 = ) i a cubic inteer 3. quare = = REMARKABLE OBSERVATIONS Let be two non-zero ditinct poitive inteer uch that p, q note that p,q p,q n n n and p q. Treat a the enerator of the Pthaorean trianle T(,,) where = pq, p q p q. Let A,P repreent the area and perimeter of T (,,). Then the followin interetin relation are oberved: = 5. A n n P 3. B coniderin the linear combination amon the olution of (), one ma obtain olution of different hperbola. A few eample are iven in table below: Table :Hperbola S.No, Hperbola 4, , , , , 3 4, , , , , 9 4, , , , , , http: // International Journal of Enineerin Science & Reearch Technolo [9]

4 [Pandichelvi*, 4.(7): Jul, 5] ISSN: (IOR), Publication Impact Factor: B coniderin linear combination amon the olution of (), one ma obtain olution of different parabola. A few eample are iven in table 3 below. Table 3:Parabola S.No, Parabola 4 8, 3 4 8, , , , 5 9 http: // Note :B coniderin the neative in on the RHS of (),another et of -value for the ame -value are obtained. PATTERN: Treatin () a a quadratic in and olvin for, we have (5) ( 4 3) Let ( 4 3) X () where X (a) The initial olution of () i, X To find the other olution of (), conider the Pellian equation X (7) whoe eneral olution i iven b where ~ ~ f, X f (5 (5 ) ) (5 (5 ) ), =,,,3... ~ X, ~ Applin Brahmaupta lemma between ( X, ) and ( ) the other inteer olution of () are iven b X f Subtitutin the above value of X, in(a),(5) and takin the poitive in on the RHS of(5), we et f f 3 The recurrence relation atified b and are iven b 3,, 3 International Journal of Enineerin Science & Reearch Technolo [9]

5 [Pandichelvi*, 4.(7): Jul, 5] ISSN: (IOR), Publication Impact Factor: , Y, Y 5 Same numerical eample of and atifin () are iven in the followin table (4) Table 4: Numerical Eample A few interetin propertie between the olution are iven below. ( ) i a nat number. i a perfect quare i a cubic inteer B coniderin linear combination amon the olution of (), one ma obtain olution of different hperbola. A few eample are iven in table (5) below Table 5:Hperbola S.No, Hperbola 4, 3 8 4, , , 5, , , , , B coniderin linear combination amon the olution of (), one ma obtain olution of different parabola. A few eample are iven in table below. Table :Parabola S.No, Parabola, 3, 5 8, http: // International Journal of Enineerin Science & Reearch Technolo [93]

6 [Pandichelvi*, 4.(7): Jul, 5] ISSN: (IOR), Publication Impact Factor: Note :B coniderin the neative in on the RHS of (5),another et of -value for the ame -value are obtained. CONCLUSION In thi paper, we have preented infinitel man inteer olution for the hperbola repreented b 4 7. A the binar quadratic diophantine equation are rich in variet, one ma earch for the other choice of hperbolic and determine their inteer olution alon with uitable propertie. REFERENCES t [] Ahmet Tekcan, The pell equation (k k), World Academ of cience, Enineerin and Technolo, 8,9,(97-7). [] Ahmet Tekcan, Betw Gezer and Oman Bizim, On the inteer olution of the pell Equation t d,world Academ of cience, Enineerin and Technolo 7,, (5-5). [3] Dickon.L.E., Hitor of Theor of number, vol.:diophantine Anali, New York, Dover, 5. t [4] Gopalan M.A.,G.umathi,S.vidhalakhmi, Obervation on the hperbola 9 3, Scholar Journal of the Enineerin and Technolo, (4), Vol:(A):5-55. [5] Gopalan M.A., S.vidhalakhmi and A.Kavitha, On the interal olution of the Binarquadratic equation, Scholar Journal of the Enineerin and Technolo, 4, Vol (A), t [] Merve Gune, Solution of the pell equation (a b b), when N (, 4), Mathematica Aeterna,, Vol, No.7 (9-38). [7] Mollin.R.A and Anitha rinivaan, A Note on the neative pell equation, International Journal of Alebra,,Vol 4,No.9,99-9. [8] Saneetha.V, M.A.Gopalan and Manju Somanath, On the interal olution of the pell equation t,international journal of appield Mathematical reearch,4,vol.3, iue (58-) 3 3 [9] M.A.Gopalan, S.Vidhalakhmi, N.Thiruniraielvi, A Stud on the Hperbola 8 3, International Journal of Latet Reearch in Science and Technolo,3,(), [] M.A.Gopalan,S.Vidhalakhmi,T.R.UhaRani,S.Mallika, Obervation on 3,Beel Journal of Mathematic,,(3), [] S.Vidhalakhmi,T.R.UhaRani,S.Mallika,V,Kumari, Obervation on the neative pellian 8 5, International Journal of Applied Reearch,5,(3),8-87. [] Gopalan.M.A, and, Shanmuanadhan.P,and Vijaahankar.A, On Binar Qudratic equation 5 8 5,Acta ciencia indica,vol.xxxivm.no.4, p.83-85(8) [3] Gopalan. M.A., and., Janaki.G., Obervation on, Impact J.Sci.,Tech,Vol(3) p.4,3-48(8). [4] Gopalan,M.A, Gokila,k, and Vidhalakhmi.S, On the Diophantine Equation 4, Acta Ciencia Indica,Vol.XXXIIIM No, p.57-57,(7). [5] Gopalan,M.A., and Parvath.G., Interal point On the Hperbola 4 4, Antarctica J.Math, Vol (), 49-55,(). [] Gopalan, M.A., Vidhalakahmi,S, Sumathi.G and Lakhmi.K, Interal point on the Hperbola 4 8 4, Beel J.Math, Vol (3),59-4,Sep(). 3 [7] Gopalan, M.A., Vidhalakahmi,S, and Devibala.S, On The Diophantine Equation 4, Acta Ciencia Indica, Vol.XXXIIIM No, P.45-45(7) [8] Gopalan, M.A., Vidhalakahmi,S, Sumathi.G and Lakhmi.K, Obervation on , Diophantu J.Math. Vol.(), 3-5,(). http: // International Journal of Enineerin Science & Reearch Technolo [94]

7 [Pandichelvi*, 4.(7): Jul, 5] ISSN: (IOR), Publication Impact Factor: [9] Gopalan, M.A., Vidhalakahmi,S, and Lakhmi.K, Obervation On The Binar Quadratic Equation 3 8 3, Scholar Journal of Phic, Mathematic and Statitic. Vol.(),(Sep-Nov),4-45,(4). [] Vidhalakhmi. S, Gopalan.M.A and Lakhmi.k,,Interal Solution of the Binar Quadratic Equation 5 33, International Journal of Innovative Science Enineerin &Technolo,Vol.(),45-453, Auut(4). [] Vidhalakhmi. S, Gopalan.M.A and Uharani.T.R,,Interal Solution of the Binar Quadratic Equation 3 5, Bulletin of Mathematic and Statitic Reearch,Vol.3 8- Iue..5. http: // International Journal of Enineerin Science & Reearch Technolo [95]