METHODS FOR DEALING WITH EXAMPLES OF SECOND-ORDER CURVES IN ANALYTICAL GEOMETRY

Аннотация

This article shows how to solve non-standard examples of second-order lines. Thisarticle can bea great guide forindependent learners. Keywords:Ellipse, parabola, hyperbola, eccentricity, directrix, focal radius, focus, asymptotic. Ellipse- the locus of points, the sum of the distances from each of which to two given points F,F (focus) is aconstant valueequal to 2a. 1 2 Ellipseelements: 2a -major axis; 2b -semi-minoraxis oftheellipse; A,A ,B ,B -peaks; 1 2 1 2 F(c;0),F (-c;0)-focus; 1 2 2c -focal length c2 =a2 -b2 c e= 1 - eccentricity (c >a) . It can be considered as a numerical characteristic of the a magnitude oftheanglebetween theasymptotes. r =±(ex-a), r =±(ex+a),-focal radius (the upper sign corresponds to the right branch, 1 2 thelowersign corresponds to theleft branch) a a d :x = , d :x =- -directrix. 1 e 2 e The geometric meaning of the imaginary axis is shown in the figure by a dotted line (the distancebetween the asymptotes). The canonical equation of a hyperbola (the coordinate axes coincide with the axes of the hyperbola): x2 y2 - =1 a2 b2 Parabola - locus of points, each of which is equidistant from a given point (focus) and from agiven line(directrix): page1115 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Elements ofaparabola: OF-focal axis; O -vertex; p F ;0 -focus; 2 e=1 -eccentricity; p r = x+ , (p >0) -focal radius; 2 p d :x =- -directrix; 2 p -focal parameter. The canonical equation of a parabola (the Ox -axis coincides with the focal axis, the origin of coordinates coincides with thevertex ofthe parabola): y2 = 2px For p 0,thebranches of theparabola are directed upwards, for p 1. a 2 a 4 Hyperboledirectrixes: x-5= ± , x-5= ± , e 13 Let's makea drawing. Draw ahyperbola anditsasymptotes, mark thecenter O(5;-1) Task.2. The general equation of the curve of the second order can be reduced to the canonical one. Find the coordinates of the center, the coordinates of the vertices and foci. Writetheequations ofasymptotes and directrixes. Draw lines ongraphs, mark points. 9x2 +25y2 -18x-100y -116 =0. page1117 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Decision. We bring the equation of the curve to the canonical form, highlighting the full squares: 9x2 +25y2 -18x-100y -116 =0. (9x2 -18x)+(25y2 -100y) =116, 9(x2 -2x)+25(y2 -4y) =116, 9(x2 -2x+1)+25(y2 -4y +4) =116 +9+100, 9(x-1)2 +25(y -2)2 = 225, (x-1)2 (y-2)2 + =1 25 9 (x-1)2 (y -2)2 + =1. 52 32 Thisis theequation ofan ellipsecentered at point O(1;2) and semiaxes a =5, b=3. Vertices in points A =(1+5;2)= A(6;2), 1 1 A =(1-5;2)= A (-4;2), 2 2 A =(1;2-3)= A (1;-1), 3 3 A =(1;2+3)= A (1;5). 4 4 Ax