{"id":225966,"date":"2025-06-04T19:10:38","date_gmt":"2025-06-04T19:10:38","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=225966"},"modified":"2025-06-04T19:10:41","modified_gmt":"2025-06-04T19:10:41","slug":"equations-of-hyperbolas-quiz-active-1-2-3-5-6-7-the-graph-of-a-hyperbola-is-shown","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/04\/equations-of-hyperbolas-quiz-active-1-2-3-5-6-7-the-graph-of-a-hyperbola-is-shown\/","title":{"rendered":"Equations of Hyperbolas Quiz Active 1 2 3 5 6 7 The graph of a hyperbola is shown"},"content":{"rendered":"\n<p>Equations of Hyperbolas Quiz Active 1 2 3 5 6 7 The graph of a hyperbola is shown. (-13, 0) (-12, 0) -21 -18 -15 -12 -9 -6 -3 Mark this and return (0, 5) 5 4 3 2 1 (12, 0) (13, 0) 3 6 9 12 15 18 21 x -1 -2 -3 -4 -5 (0, -5) -6 What are the coordinates of a focus of the hyperbola? (-12, 0) (0, -5) (0, 0) (13, 0) Save and Exit Next Submit<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p><strong>Correct Answer: (13, 0)<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h3 class=\"wp-block-heading\">Explanation<\/h3>\n\n\n\n<p>To find the coordinates of a <strong>focus<\/strong> of the hyperbola, we begin by analyzing its graph. From the visual and listed coordinates, we observe the following key points:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The <strong>vertices<\/strong> of the hyperbola are at <strong>(-13, 0)<\/strong> and <strong>(13, 0)<\/strong>.<\/li>\n\n\n\n<li>The <strong>center<\/strong> of the hyperbola is the midpoint of the segment joining the two vertices, which is <strong>(0, 0)<\/strong>.<\/li>\n\n\n\n<li>The points <strong>(0, 5)<\/strong> and <strong>(0, -5)<\/strong> lie on the conjugate axis, indicating the height of the rectangle that guides the asymptotes.<\/li>\n<\/ul>\n\n\n\n<p>This is a <strong>horizontal hyperbola<\/strong>, as the transverse axis (the line passing through the vertices) is horizontal.<\/p>\n\n\n\n<p>The standard form of a horizontal hyperbola centered at (h,k)(h, k) is: (x\u2212h)2a2\u2212(y\u2212k)2b2=1\\frac{(x &#8211; h)^2}{a^2} &#8211; \\frac{(y &#8211; k)^2}{b^2} = 1<\/p>\n\n\n\n<p>Given:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Center: (h,k)=(0,0)(h, k) = (0, 0)<\/li>\n\n\n\n<li>Distance from center to each vertex: a=13a = 13<\/li>\n\n\n\n<li>Distance from center to each co-vertex: b=5b = 5<\/li>\n<\/ul>\n\n\n\n<p>So the equation becomes: x2132\u2212y252=1\u21d2x2169\u2212y225=1\\frac{x^2}{13^2} &#8211; \\frac{y^2}{5^2} = 1 \\quad \\Rightarrow \\quad \\frac{x^2}{169} &#8211; \\frac{y^2}{25} = 1<\/p>\n\n\n\n<p>To find the coordinates of the <strong>foci<\/strong>, we use the relationship: c2=a2+b2c^2 = a^2 + b^2<\/p>\n\n\n\n<p>Substitute known values: c2=169+25=194\u21d2c=194\u224813.93c^2 = 169 + 25 = 194 \\Rightarrow c = \\sqrt{194} \\approx 13.93<\/p>\n\n\n\n<p>This tells us the foci are approximately <strong>13.93 units<\/strong> left and right of the center along the x-axis.<\/p>\n\n\n\n<p>So the coordinates of the foci are approximately: (\u00b1194,0)\u2248(\u00b113.93,0)(\\pm \\sqrt{194}, 0) \\approx (\\pm 13.93, 0)<\/p>\n\n\n\n<p>The point closest to that from the options given is <strong>(13, 0)<\/strong>.<\/p>\n\n\n\n<p>Hence, the <strong>correct answer is (13, 0)<\/strong> \u2014 one of the two foci of the hyperbola.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/06\/learnexams-banner4-382.jpeg\" alt=\"\" class=\"wp-image-225967\"\/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Equations of Hyperbolas Quiz Active 1 2 3 5 6 7 The graph of a hyperbola is shown. (-13, 0) (-12, 0) -21 -18 -15 -12 -9 -6 -3 Mark this and return (0, 5) 5 4 3 2 1 (12, 0) (13, 0) 3 6 9 12 15 18 21 x -1 -2 -3 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-225966","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225966","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=225966"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/225966\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=225966"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=225966"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=225966"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}