{"id":237167,"date":"2025-06-17T04:23:20","date_gmt":"2025-06-17T04:23:20","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=237167"},"modified":"2025-06-17T04:23:22","modified_gmt":"2025-06-17T04:23:22","slug":"the-shadow-cast-by-a-lampshade-has-a-boundary-in-the-shape-of-a-hyperbola-defined-by-the-equation-y%e2%88%92322%e2%88%92x211","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/17\/the-shadow-cast-by-a-lampshade-has-a-boundary-in-the-shape-of-a-hyperbola-defined-by-the-equation-y%e2%88%92322%e2%88%92x211\/","title":{"rendered":"The shadow cast by a lampshade has a boundary in the shape of a hyperbola defined by the equation\u00a0(y\u22123)22\u2212x21=1"},"content":{"rendered":"\n

The shadow cast by a lampshade has a boundary in the shape of a hyperbola defined by the equation (y\u22123)22\u2212x21=1<\/p>\n\n\n\n

. What could the center of the hyperbola represent? a. the location of the light bulb b. the bottom of the lampshade c. the top of the lampshade d. a vertex of the shadow<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Correct Answer: a. the location of the light bulb<\/strong><\/p>\n\n\n\n

\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The given equation of the hyperbola is: (y\u22123)222\u2212x212=1\\frac{(y – 3)^2}{2^2} – \\frac{x^2}{1^2} = 122(y\u22123)2\u200b\u221212×2\u200b=1<\/p>\n\n\n\n

This is in the standard form of a vertical hyperbola<\/strong>: (y\u2212k)2a2\u2212(x\u2212h)2b2=1\\frac{(y – k)^2}{a^2} – \\frac{(x – h)^2}{b^2} = 1a2(y\u2212k)2\u200b\u2212b2(x\u2212h)2\u200b=1<\/p>\n\n\n\n

From this, we can identify the center of the hyperbola as the point (h, k)<\/strong>. Comparing the equation:<\/p>\n\n\n\n

    \n
  • h=0h = 0h=0<\/li>\n\n\n\n
  • k=3k = 3k=3<\/li>\n<\/ul>\n\n\n\n

    So, the center is at the point (0, 3)<\/strong>.<\/p>\n\n\n\n

    In the context of a lampshade casting a shadow, the shape of the shadow being a hyperbola suggests that the light is shining through or around an object and creating that curved boundary on a surface. The center of a hyperbola<\/strong> represents the midpoint between the two branches and also aligns with the origin of the diverging light rays<\/strong> that create the shadow. This is consistent with how a point source of light<\/strong>, such as a light bulb, causes objects to cast shadows with curved edges when blocked or filtered by another object like a lampshade.<\/p>\n\n\n\n

    Since the center of this hyperbola is (0, 3), it most likely indicates the position from which the light rays originate. This is exactly what a light bulb represents<\/strong> in a real-world setting.<\/p>\n\n\n\n

    The other answer choices do not align with the mathematical interpretation:<\/p>\n\n\n\n

      \n
    • The bottom or top of the lampshade (choices b and c) are not directly linked to the center of the hyperbola<\/li>\n\n\n\n
    • A vertex (choice d) lies along the axis but is not the center<\/li>\n<\/ul>\n\n\n\n

      Therefore, the most accurate interpretation is that the center of the hyperbola represents the location of the light bulb<\/strong>.<\/p>\n\n\n\n

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      The shadow cast by a lampshade has a boundary in the shape of a hyperbola defined by the equation (y\u22123)22\u2212x21=1 . What could the center of the hyperbola represent? a. the location of the light bulb b. the bottom of the lampshade c. the top of the lampshade d. a vertex of the shadow The Correct […]<\/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-237167","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237167","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=237167"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/237167\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=237167"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=237167"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=237167"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}