METHODS FOR SIMPLIFYING THE GENERAL EQUATION OF SECOND-ORDER LINES

Аннотация

This in the article second orderly of lines general equation simplification methods analysis The line is drawn type looking at equation canonical to form to bring , to fit coordinate to the children transition and to the center according to classification methods seeing Also , simplification​ in the process used algebraic and geometric methods , their application​ and results in detail will be covered . Article second orderly the lines research in doing important theoretical thebasics and practical toapplications related recommendations own insidetakes . Key words : second orderly lines , general equation ,simplification methods ,canonical shape , coordinates replace ,center find ,algebraic methods ,geometricclassification . Second orderly lines , that is cones intersections , geometry , algebra and practical in mathematics important role plays [1-5] . Their general equation quadratic , linear terms and permanent the value own​ inside takes , this and classification and analysis to do process complicates . This the lines to study facilitate for the purpose various simplification methods working issued are , they are general equation more comfortable to form to bring opportunity gives . Such methods in line coordinates change , quadratic​ the form diagonalization and equation canonical to form to bring This​ ​ techniques application​ through second orderly of lines geometric properties determination and classification becomes easier [5-6]. This in the article general equation simplification methods , their mathematician basics and practical application​ analysis willbe done. Weare asis known second orderly curve ofthelinegeneral appearance a x 2 + 2a x y + a y 2 + 2a x + 2a y + a =0 11 12 22 13 23 33 In this way, to determine and construct a second-order line given in this form, it can be classified according towhether or notit has a center. Simplify asecond-order linearequationwith asinglecenter . In this case, using parallel translation, we place the coordinate origin at the center of the second-order line. As a result, the first terms in the equation disappear. Wedirect the coordinate axes along mutually perpendicular principal directions. Since the mutual addition of directions { } { } is an invariant property, in the new coordinate system 1,0 and 0,1 directions are mutually complementary. This condition a =0 12 isequal to theequality. So inthis case theequation of thesecond-order lineis a x 2 + a y 2 + a =0(1) 11 22 33 In this equation a 0 a 0 , a the coefficient may or may not be zero. If a the 11 22 33 33 coefficient is zero, then equation (1) page392 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue07,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Ax 2 + B y 2 =0(2) If A,Bthe coefficients have different signs, this equation defines two intersecting straight lines. If thecoefficients havethe samesigns, thisequation defines a single point. If thecoefficient inequation (1)above is not equal to zero,then equation (2)a 33 Ax 2 + B y 2 =1(3) This equation determines an ellipse or a hyperbola, depending on the sign of the coefficients. Thus, asecond-order linewitha single center consists ofone ofthefollowing fourlines: 1. Ellipse 2. hyperbole; 3. two intersecting straight lines; 4. onepoint. Simplify asecond-order linearequationwithoutasinglecenter . We then orient the new ordinate axis along a non-special principal direction. We know that this direction is non-asymptotic. We take the diameter of the axis of the ordinate as the abscissa. In { } the new coordinate system, the direction of the ordinate axis is 0,1 coordinates and the equation ofthejoint diameter inthis direction a x + a y + a =0 12 22 23 Thisequation is y = 0equivalent to theequation a =0 a = 0 a 0 12 23 22 weget relationships. Furthermore d=a a -a 2 =0 11 22 12 If we take into account the equality a =0, we get: The result is a second-order equation of 11 alinewithouta single center. a y 2 + 2a x 2 + a =0 (4) 22 13 33 equation a 0is relevant. For thisline, 22 0 0 a 13 D = 0 a 0 = -a a 2 22 22 13 a 0 a 31 33 Since , if , a 0then the second-order line has no center, and if a =0 , then the second- 13 13 order linehas infinitely many centers and thecenters form astraight line. If the second-order line does not have a center, then in equation (4) above a 0, the second- 13 a order line x = - 33 intersects the abscissa axis at a point. We move the origin to this point 2a 13 andsolvetheequation a y 2 + 2a x =0 (5) 22 13 In this equation, if a the sign of the coefficient a is opposite to the sign of the coefficient, 13 22 then equation (5) page393 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue07,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai y 2 = 2 px (6) It appears. p > 0Sinceit is inthis equation, it defines aparabola. Ifa coefficient signa If the