{"id":33734,"date":"2021-05-05T13:00:24","date_gmt":"2021-05-05T13:00:24","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=33734"},"modified":"2025-07-30T21:16:57","modified_gmt":"2025-07-30T21:16:57","slug":"aequatio-hyperbolarum-eiusque-deductio","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/aequatio-hyperbolarum-eiusque-deductio\/","title":{"rendered":"\u00c6quatio Hyperbolarum eiusque Deductio"},"content":{"rendered":"<p><center><\/p>\n<h1>\u00c6quatio Hyperbolarum eiusque Deductio<\/h1>\n<p><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione definitionem geometricam hyperbol\u00e6 explorabimus, eam cum ellipsi comparabimus, et eius \u00e6quationem generalem ac canonicam deducemus.<br \/>\n   <\/em><\/p>\n<p>   <strong>Propositi Discendi:<\/strong><br \/>\n   Hac lectione peracta discipulus poterit:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Definire<\/strong> geometrice quid sit hyperbola.<\/li>\n<li><strong>Deducere<\/strong> \u00e6quationem generalem et canonicam hyperbolarum ex definitione geometrica.<\/li>\n<li><strong>Agoscere<\/strong> differentias inter ellipses et hyperbolas respectu distantiarum focalium.<\/li>\n<\/ol>\n<p>   <strong>INDEX CONTENTORUM<\/strong><br \/>\n<a href=\"#1\">Definitio Geometrica Hyperbol\u00e6<\/a><br \/>\n<a href=\"#2\">Deductio \u00c6quationis Hyperbolarum<\/a><br \/>\n<a href=\"#3\">\u00c6quatio Generalis Hyperbolarum<\/a><br \/>\n<a href=\"#4\">\u00c6quatio Canonica Hyperbolarum<\/a>\n   <\/p>\n<p>   <\/center><\/p>\n<p>   <center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/1Aearz-E3bk\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a name=\"1\"><\/a>   <\/p>\n<h2>Definitio Geometrica Hyperbol\u00e6<\/h2>\n<p style=\"text-align: justify;\">Prius \u00e6quationem ellipsium et circulorum recognovimus et invenimus eam formam habere <span class=\"katex-eq\" data-katex-display=\"false\">ax^2 + bx + cy^2 + dy + e = 0<\/span>, ubi <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> sunt quantitates a zero distinct\u00e6 eodem signo. De hoc dictum est quod si <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">b<\/span> signa contraria habent, tunc loco ellipsis habetur Hyperbola. Nihil ultra de his curvis diximus, sed nunc illud vacuum implebimus. Studium nostrum perficemus definitione geometrica hyperbol\u00e6 proposita et ex ea \u00e6quationes generales et canonicas hyperbolarum obtinebimus.<\/p>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=176s\" target=\"_blank\" rel=\"noopener\"><strong>Ex una parte, ellipsis definiri potest<\/strong><\/a> ut collectio omnium punctorum quorum summa distantiarum ad duo puncta, qu\u00e6 foci vocantur, semper eadem est. Similiter, sed e contrario, hyperbola definitur ut collectio omnium punctorum quorum differentia absoluta distantiarum ad puncta focalia est semper eadem.<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-BNQiwaq_OJs\/YJGda6VHOmI\/AAAAAAAAFDo\/edTQHWwLGGQwyGszR7c-7H74a09ASsK2gCLcBGAsYHQ\/s0\/hiperbola%2Bdefinici%25C3%25B3n%2Bgr%25C3%25A1fica.PNG\" alt=\"Hyperbola definitio geometrica\" class=\" aligncenter lazyload\" width=\"493\" height=\"340\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-BNQiwaq_OJs\/YJGda6VHOmI\/AAAAAAAAFDo\/edTQHWwLGGQwyGszR7c-7H74a09ASsK2gCLcBGAsYHQ\/s0\/hiperbola%2Bdefinici%25C3%25B3n%2Bgr%25C3%25A1fica.PNG\" alt=\"Hyperbola definitio geometrica\" class=\" aligncenter lazyload\" width=\"493\" height=\"340\" \/><\/noscript><\/p>\n<p style=\"text-align: justify;\">Id est, satisfit relatio:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">|d(f_1,P) - d(f_2,P)| = 2a<\/span>\n<p style=\"text-align: justify;\">Ubi <span class=\"katex-eq\" data-katex-display=\"false\">a<\/span> est qu\u00e6libet quantitas realis fixa.<\/p>\n<p style=\"text-align: justify;\">Hoc revera duas \u00e6quationes generat, scilicet: <span class=\"katex-eq\" data-katex-display=\"false\">d(f_1,P) - d(f_2,P) = 2a<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">d(f_2,P) - d(f_1,P) = 2a<\/span>, unam pro qualibet rami hyperbol\u00e6.<\/p>\n<p><a name=\"2\"><\/a>  <\/p>\n<h2>Deductio \u00c6quationis Hyperbolarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=331s\" target=\"_blank\" rel=\"noopener\"><strong>Ex definitione geometrica elici potest<\/strong><\/a> repraesentatio algebraica hyperbolarum; hoc propositum ex casu simplicissimo incipiemus atque inde generalizationes extendemus. Ratiocinatio nostra pro una tantum ramorum hyperbol\u00e6 fiet; pro altera rami analogum est argumentum.<\/p>\n<h3>Deductio Form\u00e6 Simplificat\u00e6<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=356s\" target=\"_blank\" rel=\"noopener\"><strong>Consideremus duo puncta focalia<\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">f_1 = (-c,0)<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">f_2 = (c,0).<\/span> Punctum <span class=\"katex-eq\" data-katex-display=\"false\">p = (x,y)<\/span> in hyperbola continebitur si<\/p>\n<p>   <img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-SMOUgyC1lM4\/YJGg_MIkJTI\/AAAAAAAAFDw\/6JzXOcfZi70lpvTZtbC6y26AvTQzcnWNgCLcBGAsYHQ\/s0\/hiperbola%2Bcentrada%2Ben%2Bel%2Borigen.PNG\" alt=\"Hyperbola centro in origine\" class=\"aligncenter  lazyload\" width=\"342\" height=\"288\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-SMOUgyC1lM4\/YJGg_MIkJTI\/AAAAAAAAFDw\/6JzXOcfZi70lpvTZtbC6y26AvTQzcnWNgCLcBGAsYHQ\/s0\/hiperbola%2Bcentrada%2Ben%2Bel%2Borigen.PNG\" alt=\"Hyperbola centro in origine\" class=\"aligncenter  lazyload\" width=\"342\" height=\"288\" \/><\/noscript><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x+c)^2+y^2} - \\sqrt{(x-c)^2+y^2} = 2a<\/span>\n<p>   &nbsp;<\/p>\n<p style=\"text-align: justify;\">Et hinc sequitur ratio h\u00e6c:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{(x+c)^2+y^2} - \\sqrt{(x-c)^2+y^2} = 2a<\/span><\/td>\n<td>; \u00e6quatio hyperbolarum<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{x^2 + 2xc + c^2 + y^2} - \\sqrt{x^2 - 2xc + c^2 + y^2} = 2a<\/span><\/td>\n<td>; quadratorum extensio<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\sqrt{x^2 + 2xc + c^2 + y^2} = 2a + \\sqrt{x^2 - 2xc + c^2 + y^2}<\/span><\/td>\n<td>; terminos redistribuentes<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> \\color{red}{x^2} + 2xc + \\color{purple}{c^2} + \\color{violet}{y^2} = 4a^2 + 4a\\sqrt{x^2 - 2xc + c^2 + y^2} + \\color{red}{x^2} - 2xc + \\color{purple}{c^2} + \\color{violet}{y^2}<\/span><\/td>\n<td>; quadratum utrinque elevatum<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> 2xc = 4a^2 + 4a\\sqrt{x^2 - 2xc + c^2 + y^2} - 2xc <\/span><\/td>\n<td>; terminos similes sublati<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> 4xc = 4a^2 + 4a\\sqrt{x^2 - 2xc + c^2 + y^2} <\/span><\/td>\n<td>; terminos similes ordinati<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> xc = a^2 + a\\sqrt{x^2 - 2xc + c^2 + y^2} <\/span><\/td>\n<td>; terminos similes simplicati<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> xc - a^2 = a\\sqrt{x^2 - 2xc + c^2 + y^2} <\/span><\/td>\n<td>; terminos similes simplicati<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2c^2 -2xca^2 + a^4 = a^2(x^2 - 2xc + c^2 + y^2) <\/span><\/td>\n<td>; quadratum utrinque elevatum<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2c^2 \\color{red}{-2xca^2} + a^4 = a^2x^2 \\color{red}{- 2xca^2} + a^2c^2 + a^2y^2 <\/span><\/td>\n<td>; parentesi tractati<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2c^2 + a^4 = a^2x^2 + a^2c^2 + a^2y^2 <\/span><\/td>\n<td>; terminos similes sublati<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> x^2(c^2 - a^2) - a^2y^2 = a^2c^2 - a^4 = a^2(c^2 - a^2) <\/span><\/td>\n<td>; terminos reordinati<\/td>\n<\/tr>\n<tr><\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{x^2}{a^2} - \\frac{y^2}{c^2 - a^2} = 1 <\/span><\/td>\n<td>; terminos reordinati<\/td>\n<\/tr>\n<tr><\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Ad ultimam expressionem, sicut in ellipsibus, sumitur <span class=\"katex-eq\" data-katex-display=\"false\">b^2=c^2-a^2<\/span> atque \u00e6quatio hyperbolarum obtinetur<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{ \\left(\\frac{x}{a}\\right)^2 - \\left(\\frac{y}{b}\\right)^2 = 1 }<\/span>\n<p><a name=\"3\"><\/a>     <\/p>\n<h2>\u00c6quatio Generalis Hyperbolarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=801s\" target=\"_blank\" rel=\"noopener\"><strong>Ad obtinendam \u00e6quationem generalem<\/strong><\/a> hyperbolarum, satis est eam quam modo deduximus accipere et transformationes positionis applicare<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">x\\longmapsto x-h<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">y\\longmapsto y-k<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">atque hoc modo statim obtinemus \u00e6quationem generalem hyperbolarum centro <span class=\"katex-eq\" data-katex-display=\"false\">(h,k)<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{ \\left(\\frac{x-h}{a}\\right)^2 - \\left(\\frac{y-k}{b}\\right)^2 = 1 }<\/span>\n<p><a name=\"4\"><\/a>     <\/p>\n<h2>\u00c6quatio Canonica Hyperbolarum<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=1Aearz-E3bk&amp;t=974s\" target=\"_blank\" rel=\"noopener\"><strong>Et si nunc accipiamus \u00e6quationem generalem<\/strong><\/a> hyperbolarum et eam explicemus, perveniemus ad expressionem canonicam:<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\left(\\frac{x-h}{a}\\right)^2 - \\left(\\frac{y-k}{b}\\right)^2 = 1<\/span><\/td>\n<td>; \u00c6quatio generalis hyperbolarum<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\">b^2 (x^2 - 2xh + h^2) - a^2(y^2-2ky + y^2) = a^2b^2<\/span><\/td>\n<td>; quadrata resoluta et omnia multiplicata per <span class=\"katex-eq\" data-katex-display=\"false\">a^2b^2<\/span><\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2 - 2hb^2x + h^2b^2 - a^2 y^2+ 2k a^2 y - a^2 k^2 = a^2b^2<\/span><\/td>\n<td>; parenteses resoluti<\/td>\n<\/tr>\n<tr>\n<td><span class=\"katex-eq\" data-katex-display=\"false\"> b^2 x^2 - (2hb^2) x - a^2 y^2+ (2k a^2) y - (a^2b^2 + a^2 k^2 - h^2b^2) = 0 <\/span><\/td>\n<td>; terminos similes collectos<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Hoc ultimum est expressio form\u00e6 <span class=\"katex-eq\" data-katex-display=\"false\">Ax^2+Bx + Cy^2 + Dy + E = 0,<\/span> ubi <span class=\"katex-eq\" data-katex-display=\"false\">A<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">C<\/span> sunt semper a zero distincta et signis contrariis, prout iam antea pr\u00e6vidimus cum ellipses tractaremus.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u00c6quatio Hyperbolarum eiusque Deductio Summarium: In hac lectione definitionem geometricam hyperbol\u00e6 explorabimus, eam cum ellipsi comparabimus, et eius \u00e6quationem generalem ac canonicam deducemus. Propositi Discendi: Hac lectione peracta discipulus poterit: Definire geometrice quid sit hyperbola. Deducere \u00e6quationem generalem et canonicam hyperbolarum ex definitione geometrica. Agoscere differentias inter ellipses et hyperbolas respectu distantiarum focalium. INDEX CONTENTORUM [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28988,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[1304,1298],"tags":[],"class_list":["post-33734","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-et-geometria","category-mathematica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u00c6quatio Hyperbolarum eiusque Deductio - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce omnia de hyperbolis: definitionem geometricam, deductionem \u00e6quationis generalis et canonic\u00e6, differentias cum ellipsibus, atque modum solvendi \u00e6quationes gradatim. 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