{"id":249,"date":"2025-02-13T22:45:34","date_gmt":"2025-02-13T22:45:34","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-hyperbola\/"},"modified":"2025-10-22T23:05:10","modified_gmt":"2025-10-22T23:05:10","slug":"the-hyperbola","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/the-hyperbola\/","title":{"raw":"Hyperbolas: Learn It 1","rendered":"Hyperbolas: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write equations of hyperbolas in standard form.<\/li>\r\n \t<li>Graph hyperbolas.<\/li>\r\n \t<li>Solve applied problems involving hyperbolas.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"bcc-box bcc-highlight\">Hyperbolas<\/h2>\r\nWhat do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of <strong>conic<\/strong>. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182803\/CNX_Precalc_Figure_10_02_0012.jpg\" alt=\"\" width=\"487\" height=\"281\" \/>\r\n\r\nA shock wave intersecting the ground forms a portion of a conic and results in a sonic boom.\r\n\r\nMost people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.\r\n<h3>Locating the Vertices and Foci of a Hyperbola<\/h3>\r\nIn analytic geometry, a <strong>hyperbola<\/strong> is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182805\/CNX_Precalc_Figure_10_02_0022.jpg\" alt=\"\" width=\"487\" height=\"450\" \/>\r\n\r\nLike the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the difference of the distances between [latex]\\left(x,y\\right)[\/latex] and the foci is a positive constant.\r\n\r\nNotice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the <em>difference<\/em> of two distances, whereas the ellipse is defined in terms of the <em>sum<\/em> of two distances.\r\n\r\nAs with the ellipse, every hyperbola has two <strong>axes of symmetry<\/strong>. The <strong>transverse axis<\/strong> is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The <strong>conjugate axis<\/strong> is perpendicular to the transverse axis and has the co-vertices as its endpoints. The <strong>center of a hyperbola<\/strong> is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two <strong>asymptotes<\/strong> that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The <strong>central rectangle<\/strong> of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182807\/CNX_Precalc_Figure_10_02_0032.jpg\" alt=\"\" width=\"731\" height=\"437\" \/>\r\n\r\nIn this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the <em>x<\/em>- and <em>y<\/em>-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.\r\n<dl id=\"fs-id2571756\" class=\"definition\">\r\n \t<dd id=\"fs-id2571760\"><\/dd>\r\n<\/dl>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write equations of hyperbolas in standard form.<\/li>\n<li>Graph hyperbolas.<\/li>\n<li>Solve applied problems involving hyperbolas.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"bcc-box bcc-highlight\">Hyperbolas<\/h2>\n<p>What do paths of comets, supersonic booms, ancient Grecian pillars, and natural draft cooling towers have in common? They can all be modeled by the same type of <strong>conic<\/strong>. For instance, when something moves faster than the speed of sound, a shock wave in the form of a cone is created. A portion of a conic is formed when the wave intersects the ground, resulting in a sonic boom.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182803\/CNX_Precalc_Figure_10_02_0012.jpg\" alt=\"\" width=\"487\" height=\"281\" \/><\/p>\n<p>A shock wave intersecting the ground forms a portion of a conic and results in a sonic boom.<\/p>\n<p>Most people are familiar with the sonic boom created by supersonic aircraft, but humans were breaking the sound barrier long before the first supersonic flight. The crack of a whip occurs because the tip is exceeding the speed of sound. The bullets shot from many firearms also break the sound barrier, although the bang of the gun usually supersedes the sound of the sonic boom.<\/p>\n<h3>Locating the Vertices and Foci of a Hyperbola<\/h3>\n<p>In analytic geometry, a <strong>hyperbola<\/strong> is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. This intersection produces two separate unbounded curves that are mirror images of each other.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182805\/CNX_Precalc_Figure_10_02_0022.jpg\" alt=\"\" width=\"487\" height=\"450\" \/><\/p>\n<p>Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane such that the difference of the distances between [latex]\\left(x,y\\right)[\/latex] and the foci is a positive constant.<\/p>\n<p>Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the <em>difference<\/em> of two distances, whereas the ellipse is defined in terms of the <em>sum<\/em> of two distances.<\/p>\n<p>As with the ellipse, every hyperbola has two <strong>axes of symmetry<\/strong>. The <strong>transverse axis<\/strong> is a line segment that passes through the center of the hyperbola and has vertices as its endpoints. The foci lie on the line that contains the transverse axis. The <strong>conjugate axis<\/strong> is perpendicular to the transverse axis and has the co-vertices as its endpoints. The <strong>center of a hyperbola<\/strong> is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two <strong>asymptotes<\/strong> that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The <strong>central rectangle<\/strong> of the hyperbola is centered at the origin with sides that pass through each vertex and co-vertex; it is a useful tool for graphing the hyperbola and its asymptotes. To sketch the asymptotes of the hyperbola, simply sketch and extend the diagonals of the central rectangle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182807\/CNX_Precalc_Figure_10_02_0032.jpg\" alt=\"\" width=\"731\" height=\"437\" \/><\/p>\n<p>In this section, we will limit our discussion to hyperbolas that are positioned vertically or horizontally in the coordinate plane; the axes will either lie on or be parallel to the <em>x<\/em>&#8211; and <em>y<\/em>-axes. We will consider two cases: those that are centered at the origin, and those that are centered at a point other than the origin.<\/p>\n<dl id=\"fs-id2571756\" class=\"definition\">\n<dd id=\"fs-id2571760\"><\/dd>\n<\/dl>\n","protected":false},"author":6,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"learn_it","content_attributions":[{"type":"cc-attribution","description":"Precalculus","author":"OpenStax College","organization":"OpenStax","url":"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface","project":"","license":"cc-by","license_terms":""}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/249"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/249\/revisions"}],"predecessor-version":[{"id":4832,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/249\/revisions\/4832"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/249\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=249"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=249"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=249"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=249"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}