{"id":1868,"date":"2015-11-12T18:30:43","date_gmt":"2015-11-12T18:30:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1868"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"locating-the-vertices-and-foci-of-a-hyperbola","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/locating-the-vertices-and-foci-of-a-hyperbola\/","title":{"raw":"Locating the Vertices and Foci of a Hyperbola","rendered":"Locating the Vertices and Foci of a Hyperbola"},"content":{"raw":"

In analytic geometry, a 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.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 2.<\/b> A hyperbola[\/caption]\n\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.\n\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 difference<\/em> of two distances, whereas the ellipse is defined in terms of the sum<\/em> of two distances.\n\nAs with the ellipse, every hyperbola has two axes of symmetry<\/strong>. The 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 conjugate axis<\/strong> is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola<\/strong> is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes<\/strong> that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The 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.\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"\"Figure 3.<\/b> Key features of the hyperbola[\/caption]\n\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 x<\/em>- and 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>","rendered":"

In analytic geometry, a 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>\n

Figure 2.<\/b> A hyperbola<\/p>\n<\/div>\n

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

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 difference<\/em> of two distances, whereas the ellipse is defined in terms of the sum<\/em> of two distances.<\/p>\n

As with the ellipse, every hyperbola has two axes of symmetry<\/strong>. The 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 conjugate axis<\/strong> is perpendicular to the transverse axis and has the co-vertices as its endpoints. The center of a hyperbola<\/strong> is the midpoint of both the transverse and conjugate axes, where they intersect. Every hyperbola also has two asymptotes<\/strong> that pass through its center. As a hyperbola recedes from the center, its branches approach these asymptotes. The 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>\n

Figure 3.<\/b> Key features of the hyperbola<\/p>\n<\/div>\n

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 x<\/em>– and 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\n\t\t\t

\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
CC licensed content, Specific attribution<\/div>