{"id":3238,"date":"2022-06-09T19:04:07","date_gmt":"2022-06-09T17:04:07","guid":{"rendered":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/tecna-k-parabole-reseny-priklad\/"},"modified":"2022-06-13T13:29:03","modified_gmt":"2022-06-13T11:29:03","slug":"tecna-k-parabole-reseny-priklad","status":"publish","type":"lesson","link":"https:\/\/studijni-svet.cz\/kurzy\/courses\/matematika-k-maturite\/lekce\/tecna-k-parabole-reseny-priklad\/","title":{"rendered":"Te\u010dna k parabole \u0159e\u0161en\u00fd p\u0159\u00edklad"},"content":{"rendered":"<p><strong>Zad\u00e1n\u00ed:<\/strong> Parabola je d\u00e1na \u0159\u00eddic\u00ed p\u0159\u00edmkou x = -1 a ohniskem F[3; -1]. Napi\u0161te rovnici te\u010dny t\u00e9to paraboly v jej\u00edm bod\u011b T[9; 7].<\/p>\n<p><strong>\u0158e\u0161en\u00ed:\u00a0<\/strong><\/p>\n<p>Jedn\u00e1 se o p\u0159\u00edklad, ve kter\u00e9m dosad\u00edme sou\u0159adnice bodu <strong><em>T<\/em><\/strong> do rovnice te\u010dny paraboly. K tomu ale pot\u0159ebujeme z\u00edskat sou\u0159adnice vrcholu paraboly <em><strong>V<\/strong> <\/em>a parametr <strong><em>p<\/em><\/strong>. To v\u0161e se d\u00e1 snadno vy\u010d\u00edst z n\u00e1\u010drtku, kter\u00fd si m\u016f\u017eeme vytvo\u0159it.<\/p>\n<p>Nakresl\u00edme si soustavu sou\u0159adnic, do kter\u00e9 si zaneseme:<\/p>\n<ul>\n<li>\u0159\u00edd\u00edc\u00ed p\u0159\u00edmku, kter\u00e1 <strong>je rovnob\u011b\u017en\u00e1 s osou <em>y<\/em> a prot\u00edn\u00e1 osu <em>x<\/em> v bod\u011b <em>-1<\/em><\/strong><\/li>\n<li><strong>ohnisko<\/strong> <strong><em>F [3; -1]<\/em><\/strong><\/li>\n<li><strong>bod<\/strong> <em><strong>T [9; 7]<\/strong><\/em><\/li>\n<li>z ohniska <strong>povedeme kolmici na \u0159\u00edd\u00edc\u00ed p\u0159\u00edmku<\/strong> <em><strong>d, <\/strong><\/em>v m\u00edst\u011b, kde p\u0159\u00edmka protne osu <em><strong>d<\/strong><\/em> zakresl\u00edme bod <strong><em>D<\/em><\/strong><strong><em>\u00a0<\/em><\/strong>a <strong>v p\u016flce vzd\u00e1lenosti <em>FD<\/em> z\u00edsk\u00e1me vrchol<\/strong> paraboly <strong><em>V<\/em> \u21d2 <em>V[1; -1]<\/em><\/strong><\/li>\n<li><strong>vzd\u00e1lenost ohniska od \u0159\u00edd\u00edc\u00ed p\u0159\u00edmky<\/strong> <em><strong>d<\/strong><\/em> je rovna parametru paraboly (<strong><em>p=\u0399FD<\/em><em>\u0399<\/em><\/strong>)<\/li>\n<li>sou\u010dasn\u011b tak\u00e9 v\u00edme, \u017ee p\u0159\u00edmka <strong><em>FD<\/em><\/strong> <strong>je osou paraboly<\/strong><\/li>\n<li>na\u010drtneme si parabolu ve tvaru<strong> \u2282<\/strong>, d\u00edky \u010demu\u017e zjist\u00edme, jak\u00fd tvar bude m\u00edt jej\u00ed vrcholov\u00e1 rovnice a pust\u00edme se do numerick\u00e9ho \u0159e\u0161en\u00ed<\/li>\n<\/ul>\n<p style=\"text-align: center\">V\u00edme, \u017ee tvar na\u0161\u00ed vrcholov\u00e9 rovnice paraboly vypad\u00e1 takto:<\/p>\n<p style=\"text-align: center\">$$(y-n)^2=2p(x-m)$$<\/p>\n<p style=\"text-align: center\">dosad\u00edme do n\u00ed sou\u0159adnice vrcholu a parametr a z\u00edsk\u00e1me:<\/p>\n<p style=\"text-align: center\">$$(y+1)^2=8(x-1)$$<\/p>\n<p style=\"text-align: center\">N\u00e1sledn\u011b se ji\u017e m\u016f\u017eeme pustit do sestaven\u00ed rovnice te\u010dny, kter\u00e1 m\u00e1 tvar:<\/p>\n<p style=\"text-align: center\">$$(y_0-n)(y-n)=p(x_0-m)+p(x-m)$$<\/p>\n<p style=\"text-align: center\">dosad\u00edme sou\u0159adnice bodu <strong><em>T<\/em><\/strong> a vrcholu paraboly<strong><em> V<\/em><\/strong> a parametr <strong><em>p\u00a0<\/em><\/strong>a dostaneme:<\/p>\n<p style=\"text-align: center\">$$(7+1)(y+1)=4(9-1)+4(x-1)$$<\/p>\n<p style=\"text-align: center\">Uprav\u00edme:<\/p>\n<p style=\"text-align: center\">$$8y+8=32+4x-4$$<\/p>\n<p style=\"text-align: center\">$$-4x+8y-20=0 \/:(-4)$$<\/p>\n<p style=\"text-align: center\">$$x-2y+5=0$$<\/p>\n<p style=\"text-align: center\">Rovnice na\u0161\u00ed te\u010dny m\u00e1 tvar: $$x-2y+5=0$$.<\/p>\n","protected":false},"comment_status":"closed","ping_status":"closed","template":"","class_list":["post-3238","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>Te\u010dna k parabole \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\/tecna-k-parabole-reseny-priklad\/\" \/>\n<meta property=\"og:locale\" content=\"cs_CZ\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Te\u010dna k parabole \u0159e\u0161en\u00fd p\u0159\u00edklad | Studijni-svet.cz\" \/>\n<meta property=\"og:description\" content=\"Zad\u00e1n\u00ed: Parabola je d\u00e1na \u0159\u00eddic\u00ed p\u0159\u00edmkou x = -1 a ohniskem F[3; -1]. Napi\u0161te rovnici te\u010dny t\u00e9to paraboly v jej\u00edm bod\u011b T[9; 7]. \u0158e\u0161en\u00ed:\u00a0 Jedn\u00e1 se o p\u0159\u00edklad, ve kter\u00e9m dosad\u00edme sou\u0159adnice bodu T do rovnice te\u010dny paraboly. K tomu ale pot\u0159ebujeme z\u00edskat sou\u0159adnice vrcholu paraboly V a parametr p. To v\u0161e se d\u00e1 snadno ... 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Napi\u0161te rovnici te\u010dny t\u00e9to paraboly v jej\u00edm bod\u011b T[9; 7]. \u0158e\u0161en\u00ed:\u00a0 Jedn\u00e1 se o p\u0159\u00edklad, ve kter\u00e9m dosad\u00edme sou\u0159adnice bodu T do rovnice te\u010dny paraboly. K tomu ale pot\u0159ebujeme z\u00edskat sou\u0159adnice vrcholu paraboly V a parametr p. To v\u0161e se d\u00e1 snadno ... 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