{"id":814,"date":"2021-08-27T19:56:19","date_gmt":"2021-08-27T19:56:19","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=814"},"modified":"2022-10-26T02:55:58","modified_gmt":"2022-10-26T02:55:58","slug":"quadratic-surfaces","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/quadratic-surfaces\/","title":{"raw":"Quadric Surfaces","rendered":"Quadric Surfaces"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"os-abstract\">\r\n \t<li><span class=\"os-abstract-content\">Recognize the main features of ellipsoids, paraboloids, and hyperboloids.<\/span><\/li>\r\n \t<li><span class=\"os-abstract-content\">Use traces to draw the intersections of quadric surfaces with the coordinate planes.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 data-type=\"title\">Quadric Surfaces<\/h2>\r\n<p id=\"fs-id1163723819227\">We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces.<span style=\"font-size: 1em;\">\u00a0<\/span><\/p>\r\n\r\n<div data-type=\"note\">\r\n<div id=\"fs-id1163724036465\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\r\n<div id=\"fs-id1163724074875\" class=\"theorem ui-has-child-title\" data-type=\"note\">\r\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\r\n<div data-type=\"example\">\r\n<div class=\"textbox shaded\"><header>\r\n<h3 class=\"os-title\" style=\"text-align: center;\" data-type=\"title\"><span id=\"13\" class=\"os-title-label\" data-type=\"\">DEFINITION<\/span><\/h3>\r\n\r\n<hr \/>\r\n\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1163723748626\"><strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term99\" data-type=\"term\">Quadric surfaces<\/span><\/strong>\u00a0are the graphs of equations that can be expressed in the form<\/p>\r\n<p style=\"text-align: center;\">[latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1163723799136\">When a quadric surface intersects a coordinate plane, the trace is a conic section. Since these conic sections show up frequently in this section, we review some commonly encountered equations briefly below:<\/p>\r\n\r\n<div class=\"textbox examples\">\r\n<h3>Recall: Equations of Conic Sections<\/h3>\r\nThe equation [latex] x^2 = 4py [\/latex], where [latex] p [\/latex] is a constant, represents a <strong>parabola<\/strong>; the parabola is opening upward if [latex] p&gt;0 [\/latex] and downward if [latex] p&lt;0 [\/latex].\u00a0 Similarly, the equation [latex] y^2 = 4px [\/latex], where [latex] p [\/latex] is a constant, also represents a parabola; the parabola is opening rightward if [latex] p&gt;0 [\/latex] and leftward if [latex] p&lt;0 [\/latex].\r\n\r\nThe equation [latex] \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 [\/latex], where [latex] a \\text{ and } b [\/latex] are constants, represents an <strong>ellipse<\/strong>.\u00a0 It has a horizontal major axis if [latex] a &gt; b &gt; 0 [\/latex] and a vertical major axis if [latex] b &gt; a &gt; 0 [\/latex].\u00a0 The length of its horizontal axis is [latex] 2a [\/latex] and the length of its vertical axis is [latex] 2b [\/latex].\r\n\r\nThe equation [latex] \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 [\/latex], OR the equation\u00a0[latex] \\frac{y^2}{b^2} - \\frac{x^2}{a^2} = 1 [\/latex] where [latex] a \\text{ and } b [\/latex] are constants, represents an <strong>hyperbola<\/strong>.\u00a0 The hyperbola has two branches opening either to the right and left, in the case of the first equation, or up and down, in the case of the second equation.\r\n\r\n<\/div>\r\n<p id=\"fs-id1163723948496\">An\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term100\" data-type=\"term\">ellipsoid<\/span><\/strong>\u00a0is a surface described by an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]. Set [latex]x=0[\/latex] to see the trace of the ellipsoid in the [latex]yz[\/latex]-plane. To see the traces in the [latex]xy[\/latex]- and [latex]xz[\/latex]-planes, set [latex]z=0[\/latex] and [latex]y=0[\/latex], respectively. Notice that, if [latex]a=b[\/latex], the trace in the [latex]xy[\/latex]-plane is a circle. Similarly, if [latex]a=c[\/latex], the trace in the [latex]xz[\/latex]-plane is a circle and, if [latex]b=c[\/latex] then the trace in the [latex]yz[\/latex]-plane is a circle. A sphere, then, is an ellipsoid with [latex]a=b=c[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div data-type=\"example\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: sketching an ellipsoid<\/h3>\r\nSketch the ellipsoid [latex]\\frac{x^2}{2^2}+\\frac{y^2}{3^2}+\\frac{z^2}{5^2}=1[\/latex].\r\n\r\n[reveal-answer q=\"873465287\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"873465287\"]\r\n\r\nStart by sketching the traces. To find the trace in the [latex]xy[\/latex]-plane, set [latex]z=0[\/latex]: [latex]\\frac{x^2}{2^2}+\\frac{y^2}{3^2}=1[\/latex] (see\u00a0Figure 1). To find the other traces, first set [latex]y=0[\/latex] and then set [latex]x=0[\/latex].\r\n\r\n[caption id=\"attachment_5214\" align=\"aligncenter\" width=\"826\"]<img class=\"size-full wp-image-5214\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203650\/2.81.jpg\" alt=\"This figure has three images. The first image is an oval centered around the origin of the rectangular coordinate system. It intersects the x axis at -2 and 2. It intersects the y-axis at -3 and 3. The second image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -2 and 2 and the y-axis at -5 and 5. The third image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -3 and 3 and the y-axis at -5 and 5.\" width=\"826\" height=\"461\" \/> Figure 1. (a) This graph represents the trace of equation [latex]\\frac{x^2}{2^2}+\\frac{y^2}{3^2}=1[\/latex] in the [latex]xy[\/latex]-plane, when we set [latex]z=0[\/latex]. (b) When we set [latex]y=0[\/latex], we get the trace of the ellipsoid in the [latex]xz[\/latex]-plane, which is an ellipse. (c) When we set [latex]x=0[\/latex], we get the trace of the ellipsoid in the [latex]yz[\/latex]-plane, which is also an ellipse.[\/caption]Now that we know what traces of this solid look like, we can sketch the surface in three dimensions (Figure 2).\r\n\r\n[caption id=\"attachment_5215\" align=\"aligncenter\" width=\"422\"]<img class=\"size-full wp-image-5215\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203714\/2.82.jpg\" alt=\"This figure has two images. The first image is a vertical ellipse. There two curves drawn with dashed lines around the center horizontally and vertically to give the image a 3-dimensional shape. The second image is a solid elliptical shape with the center at the origin of the 3-dimensional coordinate system.\" width=\"422\" height=\"443\" \/> Figure 2. (a) The traces provide a framework for the surface. (b) The center of this ellipsoid is the origin.[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe trace of an ellipsoid is an ellipse in each of the coordinate planes. However, this does not have to be the case for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. For example, if a surface can be described by an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=\\frac{z}{c}[\/latex], then we call that surface an\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term101\" data-type=\"term\">elliptic paraboloid<\/span><\/strong>. The trace in the [latex]xy[\/latex]-plane is an ellipse, but the traces in the [latex]xz[\/latex]-plane and [latex]yz[\/latex]-plane are parabolas (Figure 3). Other elliptic paraboloids can have other orientations simply by interchanging the variables to give us a different variable in the linear term of the equation [latex]\\frac{x^2}{a^2}+\\frac{z^2}{c^2}=\\frac{y}{b}[\/latex] or\u00a0[latex]\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=\\frac{x}{a}[\/latex].\r\n\r\n[caption id=\"attachment_5217\" align=\"aligncenter\" width=\"292\"]<img class=\"size-full wp-image-5217\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203935\/2.83.jpg\" alt=\"This figure is the image of a surface. It is in the 3-dimensional coordinate system on top of the origin. A cross section of this surface parallel to the x y plane would be an ellipse.\" width=\"292\" height=\"369\" \/> Figure 3. This quadric surface is called an elliptic paraboloid.[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: identifying traces of quadric surfaces<\/h3>\r\nDescribe the traces of the elliptic paraboloid [latex]x^2+\\frac{y^2}{2^2}=\\frac{z}{5}[\/latex].\r\n\r\n[reveal-answer q=\"634623913\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"634623913\"]\r\n<p id=\"fs-id1163723685562\">To find the trace in the [latex]xy[\/latex]-plane, set [latex]z=0[\/latex]: [latex]x^2+\\frac{y^2}{2^2}=0[\/latex]. The trace in the plane [latex]z=0[\/latex] is simply one point, the origin. Since a single point does not tell us what the shape is, we can move up the [latex]z[\/latex]-axis to an arbitrary plane to find the shape of other traces of the figure.<\/p>\r\n<p id=\"fs-id1163724052111\">The trace in plane [latex]z=5[\/latex] is the graph of equation [latex]x^2+\\frac{y^2}{2^2}=1[\/latex], which is an ellipse. In the [latex]xz[\/latex]-plane, the equation becomes [latex]z=5x^{2}[\/latex]. The trace is a parabola in this plane and in any plane with the equation [latex]y=b[\/latex].<\/p>\r\n<p id=\"fs-id1163723759812\">In planes parallel to the [latex]yz[\/latex]-plane, the traces are also parabolas, as we can see in the following figure.<\/p>\r\n\r\n[caption id=\"attachment_5219\" align=\"aligncenter\" width=\"697\"]<img class=\"size-full wp-image-5219\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204036\/2.84.jpg\" alt=\"This figure has four images. The first image is the image of a surface. It is in the 3-dimensional coordinate system on top of the origin. A cross section of this surface parallel to the x y plane would be an ellipse. A cross section parallel to the x z plane would be a parabola. A cross section of the surface parallel to the y z plane would be a parabola. The second image is the cross section parallel to the x y plane and is an ellipse. The third image is the cross section parallel to the x z plane and is a parabola. The fourth image is the cross section parallel to the y z plane and is a parabola.\" width=\"697\" height=\"810\" \/> Figure 4. (a) The paraboloid [latex]x^2+\\frac{y^2}{2^2}=\\frac{z}{5}[\/latex]. (b) The trace in plane [latex]z=5[\/latex]. (c) The trace in the [latex]xz[\/latex]-plane. (d) The trace in the [latex]yz[\/latex]-plane.[\/caption][\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nA hyperboloid of one sheet is any surface that can be described with an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]. Describe the traces of the hyperboloid of one sheet given by equation [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].\r\n\r\n[reveal-answer q=\"785629873\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"785629873\"]\r\n<p id=\"fs-id1163724104708\">The traces parallel to the [latex]xy[\/latex]-plane are ellipses and the traces parallel to the [latex]xz[\/latex]- and [latex]yz[\/latex]-planes are hyperbolas. Specifically, the trace in the [latex]xy[\/latex]-plane is ellipse [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}=1[\/latex], the trace in the [latex]xz[\/latex]-plane is hyperbola [latex]\\frac{x^2}{3^2}-\\frac{z^2}{5^2}=1[\/latex], and the trace in the [latex]yz[\/latex]-plane is hyperbola [latex]\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex] (see the following figure).<span data-type=\"newline\">\r\n<\/span><\/p>\r\n\r\n[caption id=\"attachment_5224\" align=\"aligncenter\" width=\"876\"]<img class=\"size-full wp-image-5224\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204619\/2.53.jpg\" alt=\"A hyperboloid of one sheet is any surface that can be described with an equation of the form\" width=\"876\" height=\"914\" \/> Figure 5. (a) The trace in the [latex]xy[\/latex]-plane. (b) The trace in the [latex]xz[\/latex]-plane. (c) The trace in the [latex]yz[\/latex]-plane. (d) The hyperboloid [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].[\/caption][\/hidden-answer]\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try IT.\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=7809454&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=qYftkr5uwzo&amp;video_target=tpm-plugin-znpykjkm-qYftkr5uwzo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.53_transcript.html\">\u201cCP 2.53\u201d here (opens in new window).<\/a><\/span><\/center>Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in\u00a0Figure 6(a). In fact, cooling towers for nuclear power plants are often constructed in the shape of a hyperboloid. The builders are able to use straight steel beams in the construction, which makes the towers very strong while using relatively little material (Figure 6(b)).\r\n\r\n[caption id=\"attachment_5221\" align=\"aligncenter\" width=\"731\"]<img class=\"size-full wp-image-5221\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204248\/2.85.jpg\" alt=\"This figure has two images. The first image is a sculpture made of parallel sticks, curved together in a circle with a hyperbolic cross section. The second image is a nuclear power plant. The towers are hyperbolic shaped.\" width=\"731\" height=\"329\" \/> Figure 6. (a) A sculpture in the shape of a hyperboloid can be constructed of straight lines. (b) Cooling towers for nuclear power plants are often built in the shape of a hyperboloid.[\/caption]\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example: chapter opener: Finding the focus of a parabolic reflector<\/h3>\r\nEnergy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (Figure 7). If the surface of a parabolic reflector is described by equation [latex]\\frac{x^2}{100}+\\frac{y^2}{100}=\\frac{z}{4}[\/latex], where is the focal point of the reflector?\r\n\r\n[caption id=\"attachment_5251\" align=\"aligncenter\" width=\"900\"]<img class=\"size-full wp-image-5251\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31223407\/2.86.jpg\" alt=\"This figure has two images. The first image is a picture of satellite dishes with parabolic reflectors. The second image is a parabolic curve on a line segment. The bottom of the curve is at point V. There is a line segment perpendicular to the other line segment through V. There is a point on this line segment labeled F. There are 3 lines from F to the parabola, intersecting at P sub 1, P sub 2, and P sub 3. There are also three vertical lines from P sub 1 to Q sub 1, from P sub 2 to Q sub 2, and from P sub 3 to Q sub 3.\" width=\"900\" height=\"272\" \/> Figure 7. Energy reflects off of the parabolic reflector and is collected at the focal point. (credit: modification of CGP Grey, Wikimedia Commons)[\/caption]\r\n\r\n[reveal-answer q=\"166409830\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"166409830\"]\r\n\r\nSince [latex]z[\/latex] is the first-power variable, the axis of the reflector corresponds to the [latex]z[\/latex]-axis. The coefficients of [latex]x^{2}[\/latex] and [latex]y^{2}[\/latex] are equal, so the cross-section of the paraboloid perpendicular to the [latex]z[\/latex]-axis is a circle. We can consider a trace in the [latex]xz[\/latex]-plane or the [latex]yz[\/latex]-plane; the result is the same. Setting [latex]y=0[\/latex], the trace is a parabola opening up along the [latex]z[\/latex]-axis, with standard equation [latex]x^{2}=4pz[\/latex], where [latex]p[\/latex] is the focal length of the parabola. In this case, this equation becomes [latex]x^2=100\\cdot\\frac{z}4=4pz[\/latex] or [latex]25=4p[\/latex]. So [latex]p[\/latex] is [latex]6.25[\/latex] m, which tells us that the focus of the paraboloid is [latex]6.25[\/latex] m up the axis from the vertex. Because the vertex of this surface is the origin, the focal point is [latex](0, 0, 6.25)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<p id=\"fs-id1163724130437\">Seventeen standard quadric surfaces can be derived from the general equation<\/p>\r\n<p style=\"text-align: center;\">[latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/p>\r\n<p id=\"fs-id1163723954966\">The following figures summarizes the most important ones.<\/p>\r\n\r\n\r\n[caption id=\"attachment_5254\" align=\"aligncenter\" width=\"952\"]<img class=\"size-full wp-image-5254\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31223619\/2.87.jpg\" alt=\"This figure is of a table with two columns and three rows. The three rows represent the first 6 quadric surfaces: ellipsoid, hyperboloid of one sheet, and hyperboloid of two sheets. The equations and traces are in the first column. The second column has the graphs of the surfaces. The ellipsoid graph is a vertical oblong round shape. The hyperboloid of one sheet is circular on the top and the bottom and narrow in the middle. The hyperboloid in two sheets has two parabolic domes opposite of each other.\" width=\"952\" height=\"1197\" \/> Figure 8. Characteristics of Common Quadratic Surfaces: Ellipsoid, Hyperboloid of One Sheet, Hyperboloid of Two Sheets.[\/caption]\r\n\r\n<\/div>\r\n<div data-type=\"example\">\r\n\r\n[caption id=\"attachment_5256\" align=\"aligncenter\" width=\"952\"]<img class=\"size-full wp-image-5256\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31223651\/2.88.jpg\" alt=\"This figure is of a table with two columns and three rows. The three rows represent the second 6 quadric surfaces: elliptic cone, elliptic paraboloid, and hyperbolic paraboloid. The equations and traces are in the first column. The second column has the graphs of the surfaces. The elliptic cone has two cones touching at the points. The elliptic paraboloid is similar to a cone but oblong. The hyperbolic paraboloid has a bend in the middle similar to a saddle.\" width=\"952\" height=\"1163\" \/> Figure 9. Characteristics of Common Quadratic Surfaces: Elliptic Cone, Elliptic Paraboloid, Hyperbolic Paraboloid.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\"><\/div>\r\n<div id=\"fs-id1163724074875\" class=\"theorem ui-has-child-title\" data-type=\"note\">\r\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\r\n<div data-type=\"example\">\r\n<div class=\"textbox exercises\">\r\n<h3>Example: identifying equations of quadric surfaces<\/h3>\r\n<p id=\"fs-id1163723988223\">Identify the surfaces represented by the given equations.<\/p>\r\n\r\n<ol id=\"fs-id1163723987949\" type=\"a\">\r\n \t<li>[latex]16x^{2}+9y^{2}+16z^{2}=144[\/latex]<\/li>\r\n \t<li>[latex]9x^{2}-18x+4y^{2}+16y-36z+25=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"584625870\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"584625870\"]\r\n<ol id=\"fs-id1163723967396\" type=\"a\">\r\n \t<li style=\"text-align: left;\">The [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex] terms are all squared, and are all positive, so this is probably an ellipsoid. However, let\u2019s put the equation into the standard form for an ellipsoid just to be sure. We have<center>[latex]16x^{2}+9y^{2}+16z^{2}=144[\/latex].<\/center><span data-type=\"newline\"><span data-type=\"newline\">\r\n<span data-type=\"newline\">\r\n<\/span>Dividing through by [latex]144[\/latex] gives<span data-type=\"newline\">\r\n<\/span>\r\n<\/span><\/span><center>[latex]\\frac{x^2}9+\\frac{y^2}{16}+\\frac{z^2}9=1[\/latex].<\/center><span data-type=\"newline\">\r\n<\/span>So, this is, in fact, an ellipsoid, centered at the origin.<\/li>\r\n \t<li>We first notice that the [latex]z[\/latex] term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. We also note there are [latex]x[\/latex] terms and [latex]y[\/latex] terms that are not squared, so this quadric surface is not centered at the origin. We need to complete the square to put this equation in one of the standard forms. We have<span data-type=\"newline\">\r\n<\/span>\r\n<center>[latex]\\begin{aligned}\r\n9x^2-18x+4y^2+16y-36z+25&amp;= 0\\\\\r\n9x^2-18x+4y^2+16y+25&amp;=36z \\\\\r\n9(x^2-2x)+4(y^2+4y)+25&amp;=36z \\\\\r\n9(x^2-2x+1-1)+4(y^2+4y+4-4)+25&amp;=36z \\\\\r\n9(x-1)^2-9+4(y+2)^2-16+25&amp;=36z \\\\\r\n9(x-1)^2+4(y+2)^2&amp;=36z \\\\\r\n\\frac{(x-1)^2}4+\\frac{(y+2)^2}9&amp;=z.\r\n\\end{aligned}[\/latex]<\/center><span data-type=\"newline\">\r\n<\/span>This is an elliptic paraboloid centered at [latex](1, 2, 0)[\/latex].<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nIdentify the surface represented by equation [latex]9x^{2}+y^{2}-z^{2}+2z-10=0[\/latex].\r\n\r\n[reveal-answer q=\"752345122\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"752345122\"]\r\n\r\nHyperboloid of one sheet, centered at [latex](0, 0, 1)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div data-type=\"example\"><\/div>\r\n<div data-type=\"example\"><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Recognize the main features of ellipsoids, paraboloids, and hyperboloids.<\/span><\/li>\n<li><span class=\"os-abstract-content\">Use traces to draw the intersections of quadric surfaces with the coordinate planes.<\/span><\/li>\n<\/ul>\n<\/div>\n<h2 data-type=\"title\">Quadric Surfaces<\/h2>\n<p id=\"fs-id1163723819227\">We have learned about surfaces in three dimensions described by first-order equations; these are planes. Some other common types of surfaces can be described by second-order equations. We can view these surfaces as three-dimensional extensions of the conic sections we discussed earlier: the ellipse, the parabola, and the hyperbola. We call these graphs quadric surfaces.<span style=\"font-size: 1em;\">\u00a0<\/span><\/p>\n<div data-type=\"note\">\n<div id=\"fs-id1163724036465\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\">\n<div id=\"fs-id1163724074875\" class=\"theorem ui-has-child-title\" data-type=\"note\">\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\n<div data-type=\"example\">\n<div class=\"textbox shaded\">\n<header>\n<h3 class=\"os-title\" style=\"text-align: center;\" data-type=\"title\"><span id=\"13\" class=\"os-title-label\" data-type=\"\">DEFINITION<\/span><\/h3>\n<hr \/>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1163723748626\"><strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term99\" data-type=\"term\">Quadric surfaces<\/span><\/strong>\u00a0are the graphs of equations that can be expressed in the form<\/p>\n<p style=\"text-align: center;\">[latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1163723799136\">When a quadric surface intersects a coordinate plane, the trace is a conic section. Since these conic sections show up frequently in this section, we review some commonly encountered equations briefly below:<\/p>\n<div class=\"textbox examples\">\n<h3>Recall: Equations of Conic Sections<\/h3>\n<p>The equation [latex]x^2 = 4py[\/latex], where [latex]p[\/latex] is a constant, represents a <strong>parabola<\/strong>; the parabola is opening upward if [latex]p>0[\/latex] and downward if [latex]p<0[\/latex].\u00a0 Similarly, the equation [latex]y^2 = 4px[\/latex], where [latex]p[\/latex] is a constant, also represents a parabola; the parabola is opening rightward if [latex]p>0[\/latex] and leftward if [latex]p<0[\/latex].\n\nThe equation [latex]\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1[\/latex], where [latex]a \\text{ and } b[\/latex] are constants, represents an <strong>ellipse<\/strong>.\u00a0 It has a horizontal major axis if [latex]a > b > 0[\/latex] and a vertical major axis if [latex]b > a > 0[\/latex].\u00a0 The length of its horizontal axis is [latex]2a[\/latex] and the length of its vertical axis is [latex]2b[\/latex].<\/p>\n<p>The equation [latex]\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1[\/latex], OR the equation\u00a0[latex]\\frac{y^2}{b^2} - \\frac{x^2}{a^2} = 1[\/latex] where [latex]a \\text{ and } b[\/latex] are constants, represents an <strong>hyperbola<\/strong>.\u00a0 The hyperbola has two branches opening either to the right and left, in the case of the first equation, or up and down, in the case of the second equation.<\/p>\n<\/div>\n<p id=\"fs-id1163723948496\">An\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term100\" data-type=\"term\">ellipsoid<\/span><\/strong>\u00a0is a surface described by an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=1[\/latex]. Set [latex]x=0[\/latex] to see the trace of the ellipsoid in the [latex]yz[\/latex]-plane. To see the traces in the [latex]xy[\/latex]&#8211; and [latex]xz[\/latex]-planes, set [latex]z=0[\/latex] and [latex]y=0[\/latex], respectively. Notice that, if [latex]a=b[\/latex], the trace in the [latex]xy[\/latex]-plane is a circle. Similarly, if [latex]a=c[\/latex], the trace in the [latex]xz[\/latex]-plane is a circle and, if [latex]b=c[\/latex] then the trace in the [latex]yz[\/latex]-plane is a circle. A sphere, then, is an ellipsoid with [latex]a=b=c[\/latex].<\/p>\n<\/div>\n<div data-type=\"example\">\n<div class=\"textbox exercises\">\n<h3>Example: sketching an ellipsoid<\/h3>\n<p>Sketch the ellipsoid [latex]\\frac{x^2}{2^2}+\\frac{y^2}{3^2}+\\frac{z^2}{5^2}=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q873465287\">Show Solution<\/span><\/p>\n<div id=\"q873465287\" class=\"hidden-answer\" style=\"display: none\">\n<p>Start by sketching the traces. To find the trace in the [latex]xy[\/latex]-plane, set [latex]z=0[\/latex]: [latex]\\frac{x^2}{2^2}+\\frac{y^2}{3^2}=1[\/latex] (see\u00a0Figure 1). To find the other traces, first set [latex]y=0[\/latex] and then set [latex]x=0[\/latex].<\/p>\n<div id=\"attachment_5214\" style=\"width: 836px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5214\" class=\"size-full wp-image-5214\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203650\/2.81.jpg\" alt=\"This figure has three images. The first image is an oval centered around the origin of the rectangular coordinate system. It intersects the x axis at -2 and 2. It intersects the y-axis at -3 and 3. The second image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -2 and 2 and the y-axis at -5 and 5. The third image is an oval centered around the origin of the rectangular coordinate system. It intersects the x-axis at -3 and 3 and the y-axis at -5 and 5.\" width=\"826\" height=\"461\" \/><\/p>\n<p id=\"caption-attachment-5214\" class=\"wp-caption-text\">Figure 1. (a) This graph represents the trace of equation [latex]\\frac{x^2}{2^2}+\\frac{y^2}{3^2}=1[\/latex] in the [latex]xy[\/latex]-plane, when we set [latex]z=0[\/latex]. (b) When we set [latex]y=0[\/latex], we get the trace of the ellipsoid in the [latex]xz[\/latex]-plane, which is an ellipse. (c) When we set [latex]x=0[\/latex], we get the trace of the ellipsoid in the [latex]yz[\/latex]-plane, which is also an ellipse.<\/p>\n<\/div>\n<p>Now that we know what traces of this solid look like, we can sketch the surface in three dimensions (Figure 2).<\/p>\n<div id=\"attachment_5215\" style=\"width: 432px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5215\" class=\"size-full wp-image-5215\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203714\/2.82.jpg\" alt=\"This figure has two images. The first image is a vertical ellipse. There two curves drawn with dashed lines around the center horizontally and vertically to give the image a 3-dimensional shape. The second image is a solid elliptical shape with the center at the origin of the 3-dimensional coordinate system.\" width=\"422\" height=\"443\" \/><\/p>\n<p id=\"caption-attachment-5215\" class=\"wp-caption-text\">Figure 2. (a) The traces provide a framework for the surface. (b) The center of this ellipsoid is the origin.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The trace of an ellipsoid is an ellipse in each of the coordinate planes. However, this does not have to be the case for all quadric surfaces. Many quadric surfaces have traces that are different kinds of conic sections, and this is usually indicated by the name of the surface. For example, if a surface can be described by an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=\\frac{z}{c}[\/latex], then we call that surface an\u00a0<strong><span id=\"c6bfd10c-fbf4-4ebe-8e50-9b5aed4fcc9b_term101\" data-type=\"term\">elliptic paraboloid<\/span><\/strong>. The trace in the [latex]xy[\/latex]-plane is an ellipse, but the traces in the [latex]xz[\/latex]-plane and [latex]yz[\/latex]-plane are parabolas (Figure 3). Other elliptic paraboloids can have other orientations simply by interchanging the variables to give us a different variable in the linear term of the equation [latex]\\frac{x^2}{a^2}+\\frac{z^2}{c^2}=\\frac{y}{b}[\/latex] or\u00a0[latex]\\frac{y^2}{b^2}+\\frac{z^2}{c^2}=\\frac{x}{a}[\/latex].<\/p>\n<div id=\"attachment_5217\" style=\"width: 302px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5217\" class=\"size-full wp-image-5217\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31203935\/2.83.jpg\" alt=\"This figure is the image of a surface. It is in the 3-dimensional coordinate system on top of the origin. A cross section of this surface parallel to the x y plane would be an ellipse.\" width=\"292\" height=\"369\" \/><\/p>\n<p id=\"caption-attachment-5217\" class=\"wp-caption-text\">Figure 3. This quadric surface is called an elliptic paraboloid.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: identifying traces of quadric surfaces<\/h3>\n<p>Describe the traces of the elliptic paraboloid [latex]x^2+\\frac{y^2}{2^2}=\\frac{z}{5}[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q634623913\">Show Solution<\/span><\/p>\n<div id=\"q634623913\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163723685562\">To find the trace in the [latex]xy[\/latex]-plane, set [latex]z=0[\/latex]: [latex]x^2+\\frac{y^2}{2^2}=0[\/latex]. The trace in the plane [latex]z=0[\/latex] is simply one point, the origin. Since a single point does not tell us what the shape is, we can move up the [latex]z[\/latex]-axis to an arbitrary plane to find the shape of other traces of the figure.<\/p>\n<p id=\"fs-id1163724052111\">The trace in plane [latex]z=5[\/latex] is the graph of equation [latex]x^2+\\frac{y^2}{2^2}=1[\/latex], which is an ellipse. In the [latex]xz[\/latex]-plane, the equation becomes [latex]z=5x^{2}[\/latex]. The trace is a parabola in this plane and in any plane with the equation [latex]y=b[\/latex].<\/p>\n<p id=\"fs-id1163723759812\">In planes parallel to the [latex]yz[\/latex]-plane, the traces are also parabolas, as we can see in the following figure.<\/p>\n<div id=\"attachment_5219\" style=\"width: 707px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5219\" class=\"size-full wp-image-5219\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204036\/2.84.jpg\" alt=\"This figure has four images. The first image is the image of a surface. It is in the 3-dimensional coordinate system on top of the origin. A cross section of this surface parallel to the x y plane would be an ellipse. A cross section parallel to the x z plane would be a parabola. A cross section of the surface parallel to the y z plane would be a parabola. The second image is the cross section parallel to the x y plane and is an ellipse. The third image is the cross section parallel to the x z plane and is a parabola. The fourth image is the cross section parallel to the y z plane and is a parabola.\" width=\"697\" height=\"810\" \/><\/p>\n<p id=\"caption-attachment-5219\" class=\"wp-caption-text\">Figure 4. (a) The paraboloid [latex]x^2+\\frac{y^2}{2^2}=\\frac{z}{5}[\/latex]. (b) The trace in plane [latex]z=5[\/latex]. (c) The trace in the [latex]xz[\/latex]-plane. (d) The trace in the [latex]yz[\/latex]-plane.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>A hyperboloid of one sheet is any surface that can be described with an equation of the form [latex]\\frac{x^2}{a^2}+\\frac{y^2}{b^2}-\\frac{z^2}{c^2}=1[\/latex]. Describe the traces of the hyperboloid of one sheet given by equation [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q785629873\">Show Solution<\/span><\/p>\n<div id=\"q785629873\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1163724104708\">The traces parallel to the [latex]xy[\/latex]-plane are ellipses and the traces parallel to the [latex]xz[\/latex]&#8211; and [latex]yz[\/latex]-planes are hyperbolas. Specifically, the trace in the [latex]xy[\/latex]-plane is ellipse [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}=1[\/latex], the trace in the [latex]xz[\/latex]-plane is hyperbola [latex]\\frac{x^2}{3^2}-\\frac{z^2}{5^2}=1[\/latex], and the trace in the [latex]yz[\/latex]-plane is hyperbola [latex]\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex] (see the following figure).<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div id=\"attachment_5224\" style=\"width: 886px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5224\" class=\"size-full wp-image-5224\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204619\/2.53.jpg\" alt=\"A hyperboloid of one sheet is any surface that can be described with an equation of the form\" width=\"876\" height=\"914\" \/><\/p>\n<p id=\"caption-attachment-5224\" class=\"wp-caption-text\">Figure 5. (a) The trace in the [latex]xy[\/latex]-plane. (b) The trace in the [latex]xz[\/latex]-plane. (c) The trace in the [latex]yz[\/latex]-plane. (d) The hyperboloid [latex]\\frac{x^2}{3^2}+\\frac{y^2}{2^2}-\\frac{z^2}{5^2}=1[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try IT.<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=7809454&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=qYftkr5uwzo&amp;video_target=tpm-plugin-znpykjkm-qYftkr5uwzo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\"><span style=\"font-size: 1rem; orphans: 1; text-align: initial;\">You can view the transcript for <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP2.53_transcript.html\">\u201cCP 2.53\u201d here (opens in new window).<\/a><\/span><\/div>\n<p>Hyperboloids of one sheet have some fascinating properties. For example, they can be constructed using straight lines, such as in the sculpture in\u00a0Figure 6(a). In fact, cooling towers for nuclear power plants are often constructed in the shape of a hyperboloid. The builders are able to use straight steel beams in the construction, which makes the towers very strong while using relatively little material (Figure 6(b)).<\/p>\n<div id=\"attachment_5221\" style=\"width: 741px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5221\" class=\"size-full wp-image-5221\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31204248\/2.85.jpg\" alt=\"This figure has two images. The first image is a sculpture made of parallel sticks, curved together in a circle with a hyperbolic cross section. The second image is a nuclear power plant. The towers are hyperbolic shaped.\" width=\"731\" height=\"329\" \/><\/p>\n<p id=\"caption-attachment-5221\" class=\"wp-caption-text\">Figure 6. (a) A sculpture in the shape of a hyperboloid can be constructed of straight lines. (b) Cooling towers for nuclear power plants are often built in the shape of a hyperboloid.<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: chapter opener: Finding the focus of a parabolic reflector<\/h3>\n<p>Energy hitting the surface of a parabolic reflector is concentrated at the focal point of the reflector (Figure 7). If the surface of a parabolic reflector is described by equation [latex]\\frac{x^2}{100}+\\frac{y^2}{100}=\\frac{z}{4}[\/latex], where is the focal point of the reflector?<\/p>\n<div id=\"attachment_5251\" style=\"width: 910px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5251\" class=\"size-full wp-image-5251\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31223407\/2.86.jpg\" alt=\"This figure has two images. The first image is a picture of satellite dishes with parabolic reflectors. The second image is a parabolic curve on a line segment. The bottom of the curve is at point V. There is a line segment perpendicular to the other line segment through V. There is a point on this line segment labeled F. There are 3 lines from F to the parabola, intersecting at P sub 1, P sub 2, and P sub 3. There are also three vertical lines from P sub 1 to Q sub 1, from P sub 2 to Q sub 2, and from P sub 3 to Q sub 3.\" width=\"900\" height=\"272\" \/><\/p>\n<p id=\"caption-attachment-5251\" class=\"wp-caption-text\">Figure 7. Energy reflects off of the parabolic reflector and is collected at the focal point. (credit: modification of CGP Grey, Wikimedia Commons)<\/p>\n<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q166409830\">Show Solution<\/span><\/p>\n<div id=\"q166409830\" class=\"hidden-answer\" style=\"display: none\">\n<p>Since [latex]z[\/latex] is the first-power variable, the axis of the reflector corresponds to the [latex]z[\/latex]-axis. The coefficients of [latex]x^{2}[\/latex] and [latex]y^{2}[\/latex] are equal, so the cross-section of the paraboloid perpendicular to the [latex]z[\/latex]-axis is a circle. We can consider a trace in the [latex]xz[\/latex]-plane or the [latex]yz[\/latex]-plane; the result is the same. Setting [latex]y=0[\/latex], the trace is a parabola opening up along the [latex]z[\/latex]-axis, with standard equation [latex]x^{2}=4pz[\/latex], where [latex]p[\/latex] is the focal length of the parabola. In this case, this equation becomes [latex]x^2=100\\cdot\\frac{z}4=4pz[\/latex] or [latex]25=4p[\/latex]. So [latex]p[\/latex] is [latex]6.25[\/latex] m, which tells us that the focus of the paraboloid is [latex]6.25[\/latex] m up the axis from the vertex. Because the vertex of this surface is the origin, the focal point is [latex](0, 0, 6.25)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163724130437\">Seventeen standard quadric surfaces can be derived from the general equation<\/p>\n<p style=\"text-align: center;\">[latex]Ax^2+By^2+Cz^2+Dxy+Exz+Fyz+Gx+Hy+Jz+K=0[\/latex].<\/p>\n<p id=\"fs-id1163723954966\">The following figures summarizes the most important ones.<\/p>\n<div id=\"attachment_5254\" style=\"width: 962px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5254\" class=\"size-full wp-image-5254\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31223619\/2.87.jpg\" alt=\"This figure is of a table with two columns and three rows. The three rows represent the first 6 quadric surfaces: ellipsoid, hyperboloid of one sheet, and hyperboloid of two sheets. The equations and traces are in the first column. The second column has the graphs of the surfaces. The ellipsoid graph is a vertical oblong round shape. The hyperboloid of one sheet is circular on the top and the bottom and narrow in the middle. The hyperboloid in two sheets has two parabolic domes opposite of each other.\" width=\"952\" height=\"1197\" \/><\/p>\n<p id=\"caption-attachment-5254\" class=\"wp-caption-text\">Figure 8. Characteristics of Common Quadratic Surfaces: Ellipsoid, Hyperboloid of One Sheet, Hyperboloid of Two Sheets.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"example\">\n<div id=\"attachment_5256\" style=\"width: 962px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-5256\" class=\"size-full wp-image-5256\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/08\/31223651\/2.88.jpg\" alt=\"This figure is of a table with two columns and three rows. The three rows represent the second 6 quadric surfaces: elliptic cone, elliptic paraboloid, and hyperbolic paraboloid. The equations and traces are in the first column. The second column has the graphs of the surfaces. The elliptic cone has two cones touching at the points. The elliptic paraboloid is similar to a cone but oblong. The hyperbolic paraboloid has a bend in the middle similar to a saddle.\" width=\"952\" height=\"1163\" \/><\/p>\n<p id=\"caption-attachment-5256\" class=\"wp-caption-text\">Figure 9. Characteristics of Common Quadratic Surfaces: Elliptic Cone, Elliptic Paraboloid, Hyperbolic Paraboloid.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\"><\/div>\n<div id=\"fs-id1163724074875\" class=\"theorem ui-has-child-title\" data-type=\"note\">\n<div id=\"fs-id1163724080388\" class=\"ui-has-child-title\" data-type=\"example\">\n<div data-type=\"example\">\n<div class=\"textbox exercises\">\n<h3>Example: identifying equations of quadric surfaces<\/h3>\n<p id=\"fs-id1163723988223\">Identify the surfaces represented by the given equations.<\/p>\n<ol id=\"fs-id1163723987949\" type=\"a\">\n<li>[latex]16x^{2}+9y^{2}+16z^{2}=144[\/latex]<\/li>\n<li>[latex]9x^{2}-18x+4y^{2}+16y-36z+25=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q584625870\">Show Solution<\/span><\/p>\n<div id=\"q584625870\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1163723967396\" type=\"a\">\n<li style=\"text-align: left;\">The [latex]x[\/latex], [latex]y[\/latex], and [latex]z[\/latex] terms are all squared, and are all positive, so this is probably an ellipsoid. However, let\u2019s put the equation into the standard form for an ellipsoid just to be sure. We have\n<div style=\"text-align: center;\">[latex]16x^{2}+9y^{2}+16z^{2}=144[\/latex].<\/div>\n<p><span data-type=\"newline\"><span data-type=\"newline\"><br \/>\n<span data-type=\"newline\"><br \/>\n<\/span>Dividing through by [latex]144[\/latex] gives<span data-type=\"newline\"><br \/>\n<\/span><br \/>\n<\/span><\/span><\/p>\n<div style=\"text-align: center;\">[latex]\\frac{x^2}9+\\frac{y^2}{16}+\\frac{z^2}9=1[\/latex].<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>So, this is, in fact, an ellipsoid, centered at the origin.<\/li>\n<li>We first notice that the [latex]z[\/latex] term is raised only to the first power, so this is either an elliptic paraboloid or a hyperbolic paraboloid. We also note there are [latex]x[\/latex] terms and [latex]y[\/latex] terms that are not squared, so this quadric surface is not centered at the origin. We need to complete the square to put this equation in one of the standard forms. We have<span data-type=\"newline\"><br \/>\n<\/span><\/p>\n<div style=\"text-align: center;\">[latex]\\begin{aligned}  9x^2-18x+4y^2+16y-36z+25&= 0\\\\  9x^2-18x+4y^2+16y+25&=36z \\\\  9(x^2-2x)+4(y^2+4y)+25&=36z \\\\  9(x^2-2x+1-1)+4(y^2+4y+4-4)+25&=36z \\\\  9(x-1)^2-9+4(y+2)^2-16+25&=36z \\\\  9(x-1)^2+4(y+2)^2&=36z \\\\  \\frac{(x-1)^2}4+\\frac{(y+2)^2}9&=z.  \\end{aligned}[\/latex]<\/div>\n<p><span data-type=\"newline\"><br \/>\n<\/span>This is an elliptic paraboloid centered at [latex](1, 2, 0)[\/latex].<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Identify the surface represented by equation [latex]9x^{2}+y^{2}-z^{2}+2z-10=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q752345122\">Show Solution<\/span><\/p>\n<div id=\"q752345122\" class=\"hidden-answer\" style=\"display: none\">\n<p>Hyperboloid of one sheet, centered at [latex](0, 0, 1)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\"><\/div>\n<div data-type=\"example\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-814\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 2.53. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":349141,"menu_order":26,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 2.53\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-814","chapter","type-chapter","status-publish","hentry"],"part":20,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/814","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/349141"}],"version-history":[{"count":40,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/814\/revisions"}],"predecessor-version":[{"id":6441,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/814\/revisions\/6441"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/20"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/814\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=814"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=814"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=814"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=814"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}