{"id":3150,"date":"2022-05-30T18:43:02","date_gmt":"2022-05-30T16:43:02","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/elipsa-reseny-priklad\/"},"modified":"2022-05-31T11:34:18","modified_gmt":"2022-05-31T09:34:18","slug":"elipsa-reseny-priklad","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/elipsa-reseny-priklad\/","title":{"rendered":"Elipsa \u0159e\u0161en\u00fd p\u0159\u00edklad"},"content":{"rendered":"<p><strong>Zad\u00e1n\u00ed: <\/strong><\/p>\n<p>Najd\u011bte st\u0159edovou rovnici elipsy se st\u0159edem S[2; 2], v\u00fdst\u0159ednost\u00ed e = 4 a hlavn\u00edm vrcholem A[-3; 2].<\/p>\n<p><strong>\u0158e\u0161en\u00ed:<\/strong><\/p>\n<p style=\"text-align: center\">Abychom mohli dosadit do st\u0159edov\u00e9 rovnice elipsy, mus\u00edme zn\u00e1t <strong>sou\u0159adnice st\u0159edu elipsy <em>S<\/em><\/strong><\/p>\n<p style=\"text-align: center\">a <strong>d\u00e9lku hlavn\u00ed a vedlej\u0161\u00ed poloosy <em>a <\/em><\/strong>a<em><strong> b<\/strong><\/em><\/p>\n<p style=\"text-align: center\">$$\\frac{(x-m)^2}{{a}^2}+\\frac{(y-n)^2}{{b}^2}=1$$<\/p>\n<ul>\n<li style=\"text-align: left\">Nejprve si tedy vypo\u010d\u00edt\u00e1me d\u00e9lku hlavn\u00ed poloosy <strong><em>a<\/em><\/strong><\/li>\n<\/ul>\n<p style=\"text-align: left\">Tu z\u00edsk\u00e1me z v\u00fdpo\u010dtu <strong>velikosti \u00fase\u010dky <em>SA<\/em><\/strong> podle vzorce pro v\u00fdpo\u010det vzd\u00e1lenosti dvou bod\u016f (<em><strong>a=|SA|<\/strong><\/em>)<\/p>\n<p style=\"text-align: left\">Proto\u017ee zn\u00e1me sou\u0159adnice bodu <em><strong>S<\/strong><\/em> i bodu <strong><em>A<\/em><\/strong>, jednodu\u0161e dosad\u00edme do vzorce:<\/p>\n<p>$$|AB| = \\sqrt{(b_{1} &#8211; a_{1})^{2} + (b_{2} &#8211; a_{2})^{2}}$$<\/p>\n<p>Z\u00edsk\u00e1me tedy:<\/p>\n<p>$$|SA| = \\sqrt{(a_{1} &#8211; m)^{2} + (a_{2} &#8211; n)^{2}}$$<\/p>\n<p>$$|SA| = \\sqrt{(-3-2)^{2} + (2-2)^{2}}=\\sqrt25=5$$<\/p>\n<p>$$a=5$$<\/p>\n<ul>\n<li>Nyn\u00ed se m\u016f\u017eeme vrhnout na <strong>v\u00fdpo\u010det vedlej\u0161\u00ed poloosy<em> b<\/em><\/strong><\/li>\n<\/ul>\n<p>Tu m\u016f\u017eeme snadno dopo\u010d\u00edtat<strong> ze vztahu pro hlavn\u00ed poloosu, vedlej\u0161\u00ed poloosu a excentricitu\u00a0<\/strong><\/p>\n<p>$$a^{2}=b^{2}+e^{2}$$<\/p>\n<p>$$b^{2}=a^{2}-e^{2}$$<\/p>\n<p>$$b^{2}=\\sqrt{a^{2}-e^{2}}$$<\/p>\n<p>$$b=\\sqrt{5^{2}-4^{2}}$$<\/p>\n<p>$$b=3$$<\/p>\n<ul>\n<li><strong>Nyn\u00ed ji\u017e m\u016f\u017eeme dosadit do st\u0159edov\u00e9 rovnice elipsy<\/strong><\/li>\n<\/ul>\n<p>$$\\frac{(x-m)^2}{{a}^2}+\\frac{(y-n)^2}{{b}^2}=1$$<\/p>\n<p>$$\\frac{(x-2)^2}{25}+\\frac{(y-2)^2}{9}=1$$<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-3150","lesson","type-lesson","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Elipsa \u0159e\u0161en\u00fd p\u0159\u00edklad | Studijni-svet.cz<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/elipsa-reseny-priklad\/\" \/>\n<meta property=\"og:locale\" content=\"cs_CZ\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Elipsa \u0159e\u0161en\u00fd p\u0159\u00edklad | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"Zad\u00e1n\u00ed: Najd\u011bte st\u0159edovou rovnici elipsy se st\u0159edem S[2; 2], v\u00fdst\u0159ednost\u00ed e = 4 a hlavn\u00edm vrcholem A[-3; 2]. \u0158e\u0161en\u00ed: Abychom mohli dosadit do st\u0159edov\u00e9 rovnice elipsy, mus\u00edme zn\u00e1t sou\u0159adnice st\u0159edu elipsy S a d\u00e9lku hlavn\u00ed a vedlej\u0161\u00ed poloosy a a b $$frac{(x-m)^2}{{a}^2}+frac{(y-n)^2}{{b}^2}=1$$ Nejprve si tedy vypo\u010d\u00edt\u00e1me d\u00e9lku hlavn\u00ed poloosy a Tu z\u00edsk\u00e1me z v\u00fdpo\u010dtu velikosti ... 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