{"id":5257,"date":"2024-10-14T15:03:57","date_gmt":"2024-10-14T15:03:57","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=5257"},"modified":"2024-11-21T22:35:28","modified_gmt":"2024-11-21T22:35:28","slug":"conic-sections-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/conic-sections-get-stronger\/","title":{"raw":"Conic Sections: Get Stronger","rendered":"Conic Sections: Get Stronger"},"content":{"raw":"<h2>Circles<\/h2>\r\nIn the following exercises, write the standard form of the equation of the circle with the given radius and center [latex](0, 0)[\/latex].\r\n<ol>\r\n \t<li>Radius: [latex]7[\/latex]<\/li>\r\n \t<li>Radius: [latex]\\sqrt{2}[\/latex]<\/li>\r\n<\/ol>\r\nIn the following exercises, write the standard form of the equation of the circle with the given radius and center.\r\n<ol start=\"3\">\r\n \t<li>Radius: [latex]1[\/latex], center: [latex](3, 5)[\/latex]<\/li>\r\n \t<li>Radius: [latex]2.5[\/latex], center: [latex](1.5, -3.5)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.\r\n<ol start=\"5\">\r\n \t<li>Center [latex](3, -2)[\/latex] with point [latex](3, 6)[\/latex]<\/li>\r\n \t<li>Center [latex](4, 4)[\/latex] with point [latex](2, 2)[\/latex]<\/li>\r\n<\/ol>\r\nIn the following exercises,\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>find the center and radius<\/li>\r\n \t<li>graph each circle<\/li>\r\n<\/ol>\r\n<ol start=\"7\">\r\n \t<li>[latex](x + 5)^2 + (y + 3)^2 = 1[\/latex]<\/li>\r\n \t<li>[latex](x - 4)^2 + (y + 2)^2 = 16[\/latex]<\/li>\r\n \t<li>[latex]x^2 + (y + 2)^2 = 25[\/latex]<\/li>\r\n \t<li>[latex](x - 1.5)^2 + (y + 2.5)^2 = 0.25[\/latex]<\/li>\r\n \t<li>[latex]x^2 + y^2 = 64[\/latex]<\/li>\r\n \t<li>[latex]2x^2 + 2y^2 = 8[\/latex]<\/li>\r\n<\/ol>\r\nIn the following exercises,\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>identify the center and radius<\/li>\r\n \t<li>graph each circle<\/li>\r\n<\/ol>\r\n<ol start=\"13\">\r\n \t<li>[latex]x^2 + y^2 + 2x + 6y + 9 = 0[\/latex]<\/li>\r\n \t<li>[latex]x^2 + y^2 - 4x + 10y - 7 = 0[\/latex]<\/li>\r\n \t<li>[latex]x^2 + y^2 + 6y + 5 = 0[\/latex]<\/li>\r\n \t<li>[latex]x^2 + y^2 + 4x = 0[\/latex]<\/li>\r\n<\/ol>\r\n<h2>Ellipses<\/h2>\r\nFor the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.\r\n<ol>\r\n \t<li>[latex]4x^2 + 9y^2 = 36[\/latex]<\/li>\r\n \t<li>[latex]4x^2 + 9y^2 = 1[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write the equation of an ellipse in standard form, and identify the endpoints of the major and minor axes as well as the foci.\r\n<ol start=\"3\">\r\n \t<li>[latex]\\frac{x^2}{4} + \\frac{y^2}{49} = 1[\/latex]<\/li>\r\n \t<li>[latex]x^2 + 9y^2 = 1[\/latex]<\/li>\r\n \t<li>[latex]\\frac{(x - 2)^2}{49} + \\frac{(y - 4)^2}{25} = 1[\/latex]<\/li>\r\n \t<li>[latex]\\frac{(x + 5)^2}{4} + \\frac{(y - 7)^2}{9} = 1[\/latex]<\/li>\r\n \t<li>[latex]4x^2 - 8x + 9y^2 - 72y + 112 = 0[\/latex]<\/li>\r\n \t<li>[latex]4x^2 - 24x + 36y^2 - 360y + 864 = 0[\/latex]<\/li>\r\n \t<li>[latex]4x^2 + 40x + 25y^2 - 100y + 100 = 0[\/latex]<\/li>\r\n \t<li>[latex]4x^2 + 24x + 25y^2 + 200y + 336 = 0[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the foci for the given ellipses.\r\n<ol start=\"11\">\r\n \t<li>[latex]\\frac{(x+3)^2}{25} + \\frac{(y+1)^2}{36} = 1[\/latex]<\/li>\r\n \t<li>[latex]x^2 + y^2 = 1[\/latex]<\/li>\r\n \t<li>[latex]10x^2 + y^2 + 200x = 0[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1377005\">For the following exercises, graph the given ellipses, noting center, vertices, and foci.<\/p>\r\n\r\n<ol start=\"14\">\r\n \t<li>[latex]\\frac{x^2}{16} + \\frac{y^2}{9} = 1[\/latex]<\/li>\r\n \t<li>[latex]81x^2 + 49y^2 = 1[\/latex]<\/li>\r\n \t<li>[latex]\\frac{(x+3)^2}{9} + \\frac{(y-3)^2}{9} = 1[\/latex]<\/li>\r\n \t<li>[latex]4x^2 - 8x + 16y^2 - 32y - 44 = 0[\/latex]<\/li>\r\n \t<li>[latex]x^2 + 8x + 4y^2 - 40y + 112 = 0[\/latex]<\/li>\r\n \t<li>[latex]16x^2 + 64x + 4y^2 - 8y + 4 = 0[\/latex]<\/li>\r\n \t<li>[latex]4x^2 + 16x + 4y^2 + 16y + 16 = 0[\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1327626\">For the following exercises, use the given information about the graph of each ellipse to determine its equation.<\/p>\r\n\r\n<ol start=\"21\">\r\n \t<li id=\"fs-id1327631\" data-type=\"problem\">Center at the origin, symmetric with respect to the x- and y-axes, focus at [latex](0, -2)[\/latex], and point on graph [latex](5, 0)[\/latex].<\/li>\r\n \t<li data-type=\"problem\">Center [latex](4, 2)[\/latex]; vertex [latex](9, 2)[\/latex]; one focus: [latex](4 + 2\\sqrt{6}, 2)[\/latex].<\/li>\r\n \t<li data-type=\"problem\">Center [latex](-3, 4)[\/latex]; vertex [latex](1, 4)[\/latex]; one focus: [latex](-3 + 2\\sqrt{3}, 4)[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1343632\">For the following exercises, given the graph of the ellipse, determine its equation.<\/p>\r\n\r\n<ol start=\"24\">\r\n \t<li id=\"fs-id1343636\" data-type=\"problem\"><img class=\"alignnone size-full wp-image-5995\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174223\/fa3ae12abef9d2106da27b1f4e9559ee4352e676.jpg\" alt=\"A horizontal ellipse centered at (0, 0) in the x y coordinate system with vertices at (9, 0) and (negative 9, 0) and co-vertices at (0, 3) and (0, negative 3).\" width=\"458\" height=\"328\" \/><\/li>\r\n \t<li data-type=\"problem\"><img class=\"alignnone size-full wp-image-5996\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174241\/ec2cd1e9449a329eac8cc264d70cf428a14796cb.jpg\" alt=\"A vertical ellipse tangent to the y-axis at (0, 2) in the x y coordinate system and intersecting the x-axis midway between (negative 4, 0) and (negative 3, 0) and also (negative 1, 0) and (0, 0).\" width=\"311\" height=\"309\" \/><\/li>\r\n<\/ol>\r\nReal-World Applications\r\n<ol start=\"26\">\r\n \t<li>Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of [latex]h[\/latex], the height.<\/li>\r\n \t<li>An arch has the shape of a semi-ellipse. The arch has a height of [latex]12[\/latex] feet and a span of [latex]40[\/latex] feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is [latex]6[\/latex] feet. Round to the nearest hundredth.<\/li>\r\n \t<li>A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of [latex]120[\/latex] feet, and the foci are located [latex]30[\/latex] feet from the center, find the height of the ceiling at the center.<\/li>\r\n<\/ol>\r\n<h2>Hyperbolas<\/h2>\r\nFor the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.\r\n<ol>\r\n \t<li>[latex]\\frac{x^2}{36} - \\frac{y^2}{9} = 1[\/latex]<\/li>\r\n \t<li>[latex]25x^2 - 16y^2 = 400[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.\r\n<ol start=\"3\">\r\n \t<li>[latex] \\frac{x^2}{25} - \\frac{y^2}{36} = 1 [\/latex]<\/li>\r\n \t<li>[latex] \\frac{y^2}{4} - \\frac{x^2}{81} = 1 [\/latex]<\/li>\r\n \t<li>[latex] \\frac{(x - 1)^2}{9} - \\frac{(y - 2)^2}{16} = 1 [\/latex]<\/li>\r\n \t<li>[latex] \\frac{(x - 2)^2}{49} - \\frac{(y + 7)^2}{49} = 1 [\/latex]<\/li>\r\n \t<li>[latex] -9x^2 - 54x + 9y^2 - 54y + 81 = 0 [\/latex]<\/li>\r\n \t<li>[latex] -4x^2 + 24x + 16y^2 - 128y + 156 = 0 [\/latex]<\/li>\r\n \t<li>[latex] x^2 + 2x - 100y^2 - 1000y + 2401 = 0 [\/latex]<\/li>\r\n \t<li>[latex] 4x^2 + 24x - 25y^2+200y + 464 = 0 [\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.\r\n<ol start=\"11\">\r\n \t<li>[latex] \\frac{(x - 3)^2}{5^2} - \\frac{(y + 4)^2}{2^2} = 1 [\/latex]<\/li>\r\n \t<li>[latex] 9x^2 - 18x - 16y^2 + 32y - 151 = 0 [\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.\r\n<ol start=\"13\">\r\n \t<li>[latex] \\frac{x^2}{49} - \\frac{y^2}{16} = 1 [\/latex]<\/li>\r\n \t<li>[latex] \\frac{y^2}{9} - \\frac{x^2}{25} = 1 [\/latex]<\/li>\r\n \t<li>[latex] \\frac{(y + 5)^2}{9} - \\frac{(x - 4)^2}{25} = 1 [\/latex]<\/li>\r\n \t<li>[latex] \\frac{(y - 3)^2}{9} - \\frac{(x - 3)^2}{9} = 1 [\/latex]<\/li>\r\n \t<li>[latex] 64x^2 + 128x - 9y^2 - 72y - 656 = 0 [\/latex]<\/li>\r\n \t<li>[latex] -100x^2 + 1000x +y^2 -10y - 2575= 0 [\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, given information about the graph of the hyperbola, find its equation.\r\n<ol start=\"19\">\r\n \t<li>Vertices at [latex](3, 0)[\/latex] and [latex](-3, 0)[\/latex] and one focus at [latex](5, 0)[\/latex].<\/li>\r\n \t<li>Vertices at [latex](1, 1)[\/latex] and [latex](11, 1)[\/latex] and one focus at [latex](12, 1)[\/latex].<\/li>\r\n \t<li>Center: [latex](0, 0)[\/latex]; vertex: [latex](0, -13)[\/latex]; one focus: [latex]\\left(0, \\sqrt{313}\\right)[\/latex].<\/li>\r\n \t<li>Center: [latex](4, 2)[\/latex]; vertex: [latex](9, 2)[\/latex]; one focus: [latex]\\left(4 + \\sqrt{26}, 2\\right)[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1926817\">For the following exercises, given the graph of the hyperbola, find its equation.<\/p>\r\n\r\n<ol start=\"23\">\r\n \t<li id=\"fs-id1926821\" data-type=\"problem\"><img class=\"alignnone size-full wp-image-5997\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175024\/d02446c0babad7ccf528f1e87feb27806d358223.jpg\" alt=\"A vertical hyperbola centered at (0, 0) with vertices at (0, negative 4) and (0, 4). The slant asymptotes are shown but not labeled.\" width=\"487\" height=\"378\" \/><\/li>\r\n \t<li data-type=\"problem\"><img class=\"alignnone size-full wp-image-5998\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175041\/9fc5a5ef3fe1bbc5e7d7c6f6ca3875ec2e477dea.jpg\" alt=\"A vertical hyperbola centered at (negative 1, 0) with vertices at (negative 1, negative 3) and (negative 1, 3) and foci at (negative 1, negative 3 square root of 2) and (negative 1, 3 square root of 2).\" width=\"487\" height=\"441\" \/><\/li>\r\n \t<li data-type=\"problem\"><img class=\"alignnone size-full wp-image-5999\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175056\/0c786cf7752675c80480cbdefd198cf6dc830c7f.jpg\" alt=\"A horizontal hyperbola centered at (negative 3, negative 3) with vertices at (negative 8, negative 3) and (2, negative 3) and foci at (negative 3 minus 5 square root of 2, negative 3) and (negative 3 + 5 square root of 2, negative 3).\" width=\"465\" height=\"347\" \/><\/li>\r\n<\/ol>\r\nReal-World Applications\r\n\r\nFor the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.\r\n<ol start=\"26\">\r\n \t<li>The hedge will follow the asymptotes [latex] y = x [\/latex] and [latex] y = -x [\/latex], and its closest distance to the center fountain is [latex]5[\/latex] yards.<\/li>\r\n \t<li>The hedge will follow the asymptotes [latex] y = \\frac{1}{2}x [\/latex] and [latex] y = -\\frac{1}{2}x [\/latex], and its closest distance to the center fountain is [latex]10[\/latex] yards.<\/li>\r\n \t<li>The hedge will follow the asymptotes [latex] y = \\frac{3}{4}x [\/latex] and [latex] y = -\\frac{3}{4}x [\/latex], and its closest distance to the center fountain is [latex]20[\/latex] yards.<\/li>\r\n<\/ol>\r\nFor the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.\r\n<ol start=\"29\">\r\n \t<li>The object enters along a path approximated by the line [latex] y = 2x - 2 [\/latex] and passes within [latex]0.5[\/latex] au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex] y = -2x + 2 [\/latex].<\/li>\r\n \t<li>The object enters along a path approximated by the line [latex] y = \\frac{1}{3}x - 1 [\/latex] and passes within [latex]1[\/latex] au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex] y = -\\frac{1}{3}x + 1 [\/latex].<\/li>\r\n<\/ol>\r\n<h2>Parabolas<\/h2>\r\nFor the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.\r\n<ol>\r\n \t<li>[latex] y = 4x^2 [\/latex]<\/li>\r\n \t<li>[latex] (y - 3)^2 = 8(x - 2) [\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, rewrite the given equation in standard form, and then determine the vertex ([latex] V [\/latex]), focus ([latex] F [\/latex]), and directrix ([latex] d [\/latex]) of the parabola.\r\n<ol start=\"3\">\r\n \t<li>[latex] x = 8y^2 [\/latex]<\/li>\r\n \t<li>[latex] y = -4x^2 [\/latex]<\/li>\r\n \t<li>[latex]x=36y^2[\/latex]<\/li>\r\n \t<li>[latex] (x - 1)^2 = 4(y - 1) [\/latex]<\/li>\r\n \t<li>[latex] (y - 4)^2 = 2(x + 3) [\/latex]<\/li>\r\n \t<li>[latex] (x + 4)^2 = 24(y + 1) [\/latex]<\/li>\r\n \t<li>[latex] y^2 + 12x - 6y + 21 = 0 [\/latex]<\/li>\r\n \t<li>[latex] 5x^2 - 50x - 4y + 113 = 0 [\/latex]<\/li>\r\n \t<li>[latex] x^2 - 4x + 2y - 6 = 0 [\/latex]<\/li>\r\n \t<li>[latex] 3y^2 - 4x - 6y + 23 = 0 [\/latex]<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1264733\">For the following exercises, graph the parabola, labeling the focus and the directrix.<\/p>\r\n\r\n<ol start=\"13\">\r\n \t<li>[latex] x = \\frac{1}{8}y^2 [\/latex]<\/li>\r\n \t<li>[latex] y = \\frac{1}{36}x^2 [\/latex]<\/li>\r\n \t<li>[latex] (y - 2)^2 = -\\frac{4}{3}(x + 2) [\/latex]<\/li>\r\n \t<li>[latex] -6(y + 5)^2 = 4(x - 4) [\/latex]<\/li>\r\n \t<li>[latex] x^2 + 8x + 4y + 20 = 0 [\/latex]<\/li>\r\n \t<li>[latex] y^2 - 8x + 10y + 9 = 0 [\/latex]<\/li>\r\n \t<li>[latex] y^2 + 2y - 12x + 61 = 0 [\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, find the equation of the parabola given information about its graph.\r\n<ol start=\"20\">\r\n \t<li>Vertex is [latex](0, 0)[\/latex]; directrix is [latex]y = 4[\/latex], focus is [latex](0, -4)[\/latex].<\/li>\r\n \t<li>Vertex is [latex](2, 2)[\/latex]; directrix is [latex]x = 2 - \\sqrt{2}[\/latex], focus is [latex](2 + \\sqrt{2}, 2)[\/latex].<\/li>\r\n \t<li>Vertex is [latex](\\sqrt{2}, -\\sqrt{3})[\/latex]; directrix is [latex]x = 2\\sqrt{2}[\/latex], focus is [latex](0, -\\sqrt{3})[\/latex].<\/li>\r\n<\/ol>\r\n<p id=\"fs-id1525642\">For the following exercises, determine the equation for the parabola from its graph.<\/p>\r\n\r\n<ol start=\"23\">\r\n \t<li id=\"fs-id1525646\" data-type=\"problem\"><img class=\"alignnone size-full wp-image-6000\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175604\/89612f7ab67e809da1e6cdeea15e63b461c59d68.jpg\" alt=\"This figure shows two thumbtacks stuck in a piece of paper with a slack piece of string between them. A pencil pulls the string taught and by moving around, draws an ellipse.\" width=\"487\" height=\"441\" \/><\/li>\r\n \t<li data-type=\"problem\"><img class=\"alignnone size-full wp-image-6001\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175621\/856012fa53b747ad0b6e802ade89c84de6b9ee6e.jpg\" alt=\"This is a horizontal parabola in the x y plane, opening to the right, with Vertex (negative 2, 2) and Focus (negative 31\/16, 2). The Axis of Symmetry, a horizontal line, is shown, passing through the Vertex and the Focus.\" width=\"487\" height=\"441\" \/><\/li>\r\n \t<li data-type=\"problem\"><img class=\"alignnone size-full wp-image-6002\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175636\/577e8958e39b68c03ae3b39a12400bbfe208b71d.jpg\" alt=\"This is a horizontal parabola in the x y plane, opening to the right, with Vertex (negative square root of 2, square root of 3) and Focus (negative square root of 2 + square root of 5, square root of 3). The Axis of Symmetry, a horizontal line, is shown, passing through the Vertex and the Focus.\" width=\"487\" height=\"441\" \/><\/li>\r\n<\/ol>\r\nReal-World Applications\r\n<ol start=\"26\">\r\n \t<li>The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as [latex] x^2 = 4y [\/latex]. At what coordinates should you place the light bulb?<\/li>\r\n \t<li>A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is [latex]12[\/latex] feet across at its opening and [latex]4[\/latex] feet deep at its center, where should the receiver be placed?<\/li>\r\n \t<li>The reflector in a searchlight is shaped like a paraboloid of revolution. A light source is located [latex]1[\/latex] foot from the base along the axis of symmetry. If the opening of the searchlight is [latex]3[\/latex] feet across, find the depth.<\/li>\r\n \t<li>An arch is in the shape of a parabola. It has a span of [latex]100[\/latex] feet and a maximum height of [latex]20[\/latex] feet. Find the equation of the parabola, and determine the height of the arch [latex]40[\/latex] feet from the center.<\/li>\r\n \t<li>An object is projected so as to follow a parabolic path given by [latex] y = -x^2 + 96x [\/latex], where [latex] x [\/latex] is the horizontal distance traveled in feet and [latex] y [\/latex] is the height. Determine the maximum height the object reaches.<\/li>\r\n<\/ol>","rendered":"<h2>Circles<\/h2>\n<p>In the following exercises, write the standard form of the equation of the circle with the given radius and center [latex](0, 0)[\/latex].<\/p>\n<ol>\n<li>Radius: [latex]7[\/latex]<\/li>\n<li>Radius: [latex]\\sqrt{2}[\/latex]<\/li>\n<\/ol>\n<p>In the following exercises, write the standard form of the equation of the circle with the given radius and center.<\/p>\n<ol start=\"3\">\n<li>Radius: [latex]1[\/latex], center: [latex](3, 5)[\/latex]<\/li>\n<li>Radius: [latex]2.5[\/latex], center: [latex](1.5, -3.5)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.<\/p>\n<ol start=\"5\">\n<li>Center [latex](3, -2)[\/latex] with point [latex](3, 6)[\/latex]<\/li>\n<li>Center [latex](4, 4)[\/latex] with point [latex](2, 2)[\/latex]<\/li>\n<\/ol>\n<p>In the following exercises,<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>find the center and radius<\/li>\n<li>graph each circle<\/li>\n<\/ol>\n<ol start=\"7\">\n<li>[latex](x + 5)^2 + (y + 3)^2 = 1[\/latex]<\/li>\n<li>[latex](x - 4)^2 + (y + 2)^2 = 16[\/latex]<\/li>\n<li>[latex]x^2 + (y + 2)^2 = 25[\/latex]<\/li>\n<li>[latex](x - 1.5)^2 + (y + 2.5)^2 = 0.25[\/latex]<\/li>\n<li>[latex]x^2 + y^2 = 64[\/latex]<\/li>\n<li>[latex]2x^2 + 2y^2 = 8[\/latex]<\/li>\n<\/ol>\n<p>In the following exercises,<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>identify the center and radius<\/li>\n<li>graph each circle<\/li>\n<\/ol>\n<ol start=\"13\">\n<li>[latex]x^2 + y^2 + 2x + 6y + 9 = 0[\/latex]<\/li>\n<li>[latex]x^2 + y^2 - 4x + 10y - 7 = 0[\/latex]<\/li>\n<li>[latex]x^2 + y^2 + 6y + 5 = 0[\/latex]<\/li>\n<li>[latex]x^2 + y^2 + 4x = 0[\/latex]<\/li>\n<\/ol>\n<h2>Ellipses<\/h2>\n<p>For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.<\/p>\n<ol>\n<li>[latex]4x^2 + 9y^2 = 36[\/latex]<\/li>\n<li>[latex]4x^2 + 9y^2 = 1[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write the equation of an ellipse in standard form, and identify the endpoints of the major and minor axes as well as the foci.<\/p>\n<ol start=\"3\">\n<li>[latex]\\frac{x^2}{4} + \\frac{y^2}{49} = 1[\/latex]<\/li>\n<li>[latex]x^2 + 9y^2 = 1[\/latex]<\/li>\n<li>[latex]\\frac{(x - 2)^2}{49} + \\frac{(y - 4)^2}{25} = 1[\/latex]<\/li>\n<li>[latex]\\frac{(x + 5)^2}{4} + \\frac{(y - 7)^2}{9} = 1[\/latex]<\/li>\n<li>[latex]4x^2 - 8x + 9y^2 - 72y + 112 = 0[\/latex]<\/li>\n<li>[latex]4x^2 - 24x + 36y^2 - 360y + 864 = 0[\/latex]<\/li>\n<li>[latex]4x^2 + 40x + 25y^2 - 100y + 100 = 0[\/latex]<\/li>\n<li>[latex]4x^2 + 24x + 25y^2 + 200y + 336 = 0[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, find the foci for the given ellipses.<\/p>\n<ol start=\"11\">\n<li>[latex]\\frac{(x+3)^2}{25} + \\frac{(y+1)^2}{36} = 1[\/latex]<\/li>\n<li>[latex]x^2 + y^2 = 1[\/latex]<\/li>\n<li>[latex]10x^2 + y^2 + 200x = 0[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1377005\">For the following exercises, graph the given ellipses, noting center, vertices, and foci.<\/p>\n<ol start=\"14\">\n<li>[latex]\\frac{x^2}{16} + \\frac{y^2}{9} = 1[\/latex]<\/li>\n<li>[latex]81x^2 + 49y^2 = 1[\/latex]<\/li>\n<li>[latex]\\frac{(x+3)^2}{9} + \\frac{(y-3)^2}{9} = 1[\/latex]<\/li>\n<li>[latex]4x^2 - 8x + 16y^2 - 32y - 44 = 0[\/latex]<\/li>\n<li>[latex]x^2 + 8x + 4y^2 - 40y + 112 = 0[\/latex]<\/li>\n<li>[latex]16x^2 + 64x + 4y^2 - 8y + 4 = 0[\/latex]<\/li>\n<li>[latex]4x^2 + 16x + 4y^2 + 16y + 16 = 0[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1327626\">For the following exercises, use the given information about the graph of each ellipse to determine its equation.<\/p>\n<ol start=\"21\">\n<li id=\"fs-id1327631\" data-type=\"problem\">Center at the origin, symmetric with respect to the x- and y-axes, focus at [latex](0, -2)[\/latex], and point on graph [latex](5, 0)[\/latex].<\/li>\n<li data-type=\"problem\">Center [latex](4, 2)[\/latex]; vertex [latex](9, 2)[\/latex]; one focus: [latex](4 + 2\\sqrt{6}, 2)[\/latex].<\/li>\n<li data-type=\"problem\">Center [latex](-3, 4)[\/latex]; vertex [latex](1, 4)[\/latex]; one focus: [latex](-3 + 2\\sqrt{3}, 4)[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1343632\">For the following exercises, given the graph of the ellipse, determine its equation.<\/p>\n<ol start=\"24\">\n<li id=\"fs-id1343636\" data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5995\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174223\/fa3ae12abef9d2106da27b1f4e9559ee4352e676.jpg\" alt=\"A horizontal ellipse centered at (0, 0) in the x y coordinate system with vertices at (9, 0) and (negative 9, 0) and co-vertices at (0, 3) and (0, negative 3).\" width=\"458\" height=\"328\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174223\/fa3ae12abef9d2106da27b1f4e9559ee4352e676.jpg 458w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174223\/fa3ae12abef9d2106da27b1f4e9559ee4352e676-300x215.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174223\/fa3ae12abef9d2106da27b1f4e9559ee4352e676-65x47.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174223\/fa3ae12abef9d2106da27b1f4e9559ee4352e676-225x161.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174223\/fa3ae12abef9d2106da27b1f4e9559ee4352e676-350x251.jpg 350w\" sizes=\"(max-width: 458px) 100vw, 458px\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5996\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174241\/ec2cd1e9449a329eac8cc264d70cf428a14796cb.jpg\" alt=\"A vertical ellipse tangent to the y-axis at (0, 2) in the x y coordinate system and intersecting the x-axis midway between (negative 4, 0) and (negative 3, 0) and also (negative 1, 0) and (0, 0).\" width=\"311\" height=\"309\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174241\/ec2cd1e9449a329eac8cc264d70cf428a14796cb.jpg 311w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174241\/ec2cd1e9449a329eac8cc264d70cf428a14796cb-300x298.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174241\/ec2cd1e9449a329eac8cc264d70cf428a14796cb-150x150.jpg 150w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174241\/ec2cd1e9449a329eac8cc264d70cf428a14796cb-65x65.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30174241\/ec2cd1e9449a329eac8cc264d70cf428a14796cb-225x224.jpg 225w\" sizes=\"(max-width: 311px) 100vw, 311px\" \/><\/li>\n<\/ol>\n<p>Real-World Applications<\/p>\n<ol start=\"26\">\n<li>Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of [latex]h[\/latex], the height.<\/li>\n<li>An arch has the shape of a semi-ellipse. The arch has a height of [latex]12[\/latex] feet and a span of [latex]40[\/latex] feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is [latex]6[\/latex] feet. Round to the nearest hundredth.<\/li>\n<li>A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of [latex]120[\/latex] feet, and the foci are located [latex]30[\/latex] feet from the center, find the height of the ceiling at the center.<\/li>\n<\/ol>\n<h2>Hyperbolas<\/h2>\n<p>For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.<\/p>\n<ol>\n<li>[latex]\\frac{x^2}{36} - \\frac{y^2}{9} = 1[\/latex]<\/li>\n<li>[latex]25x^2 - 16y^2 = 400[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.<\/p>\n<ol start=\"3\">\n<li>[latex]\\frac{x^2}{25} - \\frac{y^2}{36} = 1[\/latex]<\/li>\n<li>[latex]\\frac{y^2}{4} - \\frac{x^2}{81} = 1[\/latex]<\/li>\n<li>[latex]\\frac{(x - 1)^2}{9} - \\frac{(y - 2)^2}{16} = 1[\/latex]<\/li>\n<li>[latex]\\frac{(x - 2)^2}{49} - \\frac{(y + 7)^2}{49} = 1[\/latex]<\/li>\n<li>[latex]-9x^2 - 54x + 9y^2 - 54y + 81 = 0[\/latex]<\/li>\n<li>[latex]-4x^2 + 24x + 16y^2 - 128y + 156 = 0[\/latex]<\/li>\n<li>[latex]x^2 + 2x - 100y^2 - 1000y + 2401 = 0[\/latex]<\/li>\n<li>[latex]4x^2 + 24x - 25y^2+200y + 464 = 0[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.<\/p>\n<ol start=\"11\">\n<li>[latex]\\frac{(x - 3)^2}{5^2} - \\frac{(y + 4)^2}{2^2} = 1[\/latex]<\/li>\n<li>[latex]9x^2 - 18x - 16y^2 + 32y - 151 = 0[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.<\/p>\n<ol start=\"13\">\n<li>[latex]\\frac{x^2}{49} - \\frac{y^2}{16} = 1[\/latex]<\/li>\n<li>[latex]\\frac{y^2}{9} - \\frac{x^2}{25} = 1[\/latex]<\/li>\n<li>[latex]\\frac{(y + 5)^2}{9} - \\frac{(x - 4)^2}{25} = 1[\/latex]<\/li>\n<li>[latex]\\frac{(y - 3)^2}{9} - \\frac{(x - 3)^2}{9} = 1[\/latex]<\/li>\n<li>[latex]64x^2 + 128x - 9y^2 - 72y - 656 = 0[\/latex]<\/li>\n<li>[latex]-100x^2 + 1000x +y^2 -10y - 2575= 0[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, given information about the graph of the hyperbola, find its equation.<\/p>\n<ol start=\"19\">\n<li>Vertices at [latex](3, 0)[\/latex] and [latex](-3, 0)[\/latex] and one focus at [latex](5, 0)[\/latex].<\/li>\n<li>Vertices at [latex](1, 1)[\/latex] and [latex](11, 1)[\/latex] and one focus at [latex](12, 1)[\/latex].<\/li>\n<li>Center: [latex](0, 0)[\/latex]; vertex: [latex](0, -13)[\/latex]; one focus: [latex]\\left(0, \\sqrt{313}\\right)[\/latex].<\/li>\n<li>Center: [latex](4, 2)[\/latex]; vertex: [latex](9, 2)[\/latex]; one focus: [latex]\\left(4 + \\sqrt{26}, 2\\right)[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1926817\">For the following exercises, given the graph of the hyperbola, find its equation.<\/p>\n<ol start=\"23\">\n<li id=\"fs-id1926821\" data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5997\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175024\/d02446c0babad7ccf528f1e87feb27806d358223.jpg\" alt=\"A vertical hyperbola centered at (0, 0) with vertices at (0, negative 4) and (0, 4). The slant asymptotes are shown but not labeled.\" width=\"487\" height=\"378\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175024\/d02446c0babad7ccf528f1e87feb27806d358223.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175024\/d02446c0babad7ccf528f1e87feb27806d358223-300x233.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175024\/d02446c0babad7ccf528f1e87feb27806d358223-65x50.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175024\/d02446c0babad7ccf528f1e87feb27806d358223-225x175.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175024\/d02446c0babad7ccf528f1e87feb27806d358223-350x272.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5998\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175041\/9fc5a5ef3fe1bbc5e7d7c6f6ca3875ec2e477dea.jpg\" alt=\"A vertical hyperbola centered at (negative 1, 0) with vertices at (negative 1, negative 3) and (negative 1, 3) and foci at (negative 1, negative 3 square root of 2) and (negative 1, 3 square root of 2).\" width=\"487\" height=\"441\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175041\/9fc5a5ef3fe1bbc5e7d7c6f6ca3875ec2e477dea.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175041\/9fc5a5ef3fe1bbc5e7d7c6f6ca3875ec2e477dea-300x272.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175041\/9fc5a5ef3fe1bbc5e7d7c6f6ca3875ec2e477dea-65x59.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175041\/9fc5a5ef3fe1bbc5e7d7c6f6ca3875ec2e477dea-225x204.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175041\/9fc5a5ef3fe1bbc5e7d7c6f6ca3875ec2e477dea-350x317.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-5999\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175056\/0c786cf7752675c80480cbdefd198cf6dc830c7f.jpg\" alt=\"A horizontal hyperbola centered at (negative 3, negative 3) with vertices at (negative 8, negative 3) and (2, negative 3) and foci at (negative 3 minus 5 square root of 2, negative 3) and (negative 3 + 5 square root of 2, negative 3).\" width=\"465\" height=\"347\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175056\/0c786cf7752675c80480cbdefd198cf6dc830c7f.jpg 465w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175056\/0c786cf7752675c80480cbdefd198cf6dc830c7f-300x224.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175056\/0c786cf7752675c80480cbdefd198cf6dc830c7f-65x49.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175056\/0c786cf7752675c80480cbdefd198cf6dc830c7f-225x168.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175056\/0c786cf7752675c80480cbdefd198cf6dc830c7f-350x261.jpg 350w\" sizes=\"(max-width: 465px) 100vw, 465px\" \/><\/li>\n<\/ol>\n<p>Real-World Applications<\/p>\n<p>For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.<\/p>\n<ol start=\"26\">\n<li>The hedge will follow the asymptotes [latex]y = x[\/latex] and [latex]y = -x[\/latex], and its closest distance to the center fountain is [latex]5[\/latex] yards.<\/li>\n<li>The hedge will follow the asymptotes [latex]y = \\frac{1}{2}x[\/latex] and [latex]y = -\\frac{1}{2}x[\/latex], and its closest distance to the center fountain is [latex]10[\/latex] yards.<\/li>\n<li>The hedge will follow the asymptotes [latex]y = \\frac{3}{4}x[\/latex] and [latex]y = -\\frac{3}{4}x[\/latex], and its closest distance to the center fountain is [latex]20[\/latex] yards.<\/li>\n<\/ol>\n<p>For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object&#8217;s path. Give the equation of the flight path of each object using the given information.<\/p>\n<ol start=\"29\">\n<li>The object enters along a path approximated by the line [latex]y = 2x - 2[\/latex] and passes within [latex]0.5[\/latex] au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex]y = -2x + 2[\/latex].<\/li>\n<li>The object enters along a path approximated by the line [latex]y = \\frac{1}{3}x - 1[\/latex] and passes within [latex]1[\/latex] au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex]y = -\\frac{1}{3}x + 1[\/latex].<\/li>\n<\/ol>\n<h2>Parabolas<\/h2>\n<p>For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.<\/p>\n<ol>\n<li>[latex]y = 4x^2[\/latex]<\/li>\n<li>[latex](y - 3)^2 = 8(x - 2)[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, rewrite the given equation in standard form, and then determine the vertex ([latex]V[\/latex]), focus ([latex]F[\/latex]), and directrix ([latex]d[\/latex]) of the parabola.<\/p>\n<ol start=\"3\">\n<li>[latex]x = 8y^2[\/latex]<\/li>\n<li>[latex]y = -4x^2[\/latex]<\/li>\n<li>[latex]x=36y^2[\/latex]<\/li>\n<li>[latex](x - 1)^2 = 4(y - 1)[\/latex]<\/li>\n<li>[latex](y - 4)^2 = 2(x + 3)[\/latex]<\/li>\n<li>[latex](x + 4)^2 = 24(y + 1)[\/latex]<\/li>\n<li>[latex]y^2 + 12x - 6y + 21 = 0[\/latex]<\/li>\n<li>[latex]5x^2 - 50x - 4y + 113 = 0[\/latex]<\/li>\n<li>[latex]x^2 - 4x + 2y - 6 = 0[\/latex]<\/li>\n<li>[latex]3y^2 - 4x - 6y + 23 = 0[\/latex]<\/li>\n<\/ol>\n<p id=\"fs-id1264733\">For the following exercises, graph the parabola, labeling the focus and the directrix.<\/p>\n<ol start=\"13\">\n<li>[latex]x = \\frac{1}{8}y^2[\/latex]<\/li>\n<li>[latex]y = \\frac{1}{36}x^2[\/latex]<\/li>\n<li>[latex](y - 2)^2 = -\\frac{4}{3}(x + 2)[\/latex]<\/li>\n<li>[latex]-6(y + 5)^2 = 4(x - 4)[\/latex]<\/li>\n<li>[latex]x^2 + 8x + 4y + 20 = 0[\/latex]<\/li>\n<li>[latex]y^2 - 8x + 10y + 9 = 0[\/latex]<\/li>\n<li>[latex]y^2 + 2y - 12x + 61 = 0[\/latex]<\/li>\n<\/ol>\n<p>For the following exercises, find the equation of the parabola given information about its graph.<\/p>\n<ol start=\"20\">\n<li>Vertex is [latex](0, 0)[\/latex]; directrix is [latex]y = 4[\/latex], focus is [latex](0, -4)[\/latex].<\/li>\n<li>Vertex is [latex](2, 2)[\/latex]; directrix is [latex]x = 2 - \\sqrt{2}[\/latex], focus is [latex](2 + \\sqrt{2}, 2)[\/latex].<\/li>\n<li>Vertex is [latex](\\sqrt{2}, -\\sqrt{3})[\/latex]; directrix is [latex]x = 2\\sqrt{2}[\/latex], focus is [latex](0, -\\sqrt{3})[\/latex].<\/li>\n<\/ol>\n<p id=\"fs-id1525642\">For the following exercises, determine the equation for the parabola from its graph.<\/p>\n<ol start=\"23\">\n<li id=\"fs-id1525646\" data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6000\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175604\/89612f7ab67e809da1e6cdeea15e63b461c59d68.jpg\" alt=\"This figure shows two thumbtacks stuck in a piece of paper with a slack piece of string between them. A pencil pulls the string taught and by moving around, draws an ellipse.\" width=\"487\" height=\"441\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175604\/89612f7ab67e809da1e6cdeea15e63b461c59d68.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175604\/89612f7ab67e809da1e6cdeea15e63b461c59d68-300x272.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175604\/89612f7ab67e809da1e6cdeea15e63b461c59d68-65x59.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175604\/89612f7ab67e809da1e6cdeea15e63b461c59d68-225x204.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175604\/89612f7ab67e809da1e6cdeea15e63b461c59d68-350x317.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6001\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175621\/856012fa53b747ad0b6e802ade89c84de6b9ee6e.jpg\" alt=\"This is a horizontal parabola in the x y plane, opening to the right, with Vertex (negative 2, 2) and Focus (negative 31\/16, 2). The Axis of Symmetry, a horizontal line, is shown, passing through the Vertex and the Focus.\" width=\"487\" height=\"441\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175621\/856012fa53b747ad0b6e802ade89c84de6b9ee6e.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175621\/856012fa53b747ad0b6e802ade89c84de6b9ee6e-300x272.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175621\/856012fa53b747ad0b6e802ade89c84de6b9ee6e-65x59.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175621\/856012fa53b747ad0b6e802ade89c84de6b9ee6e-225x204.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175621\/856012fa53b747ad0b6e802ade89c84de6b9ee6e-350x317.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/li>\n<li data-type=\"problem\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6002\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175636\/577e8958e39b68c03ae3b39a12400bbfe208b71d.jpg\" alt=\"This is a horizontal parabola in the x y plane, opening to the right, with Vertex (negative square root of 2, square root of 3) and Focus (negative square root of 2 + square root of 5, square root of 3). The Axis of Symmetry, a horizontal line, is shown, passing through the Vertex and the Focus.\" width=\"487\" height=\"441\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175636\/577e8958e39b68c03ae3b39a12400bbfe208b71d.jpg 487w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175636\/577e8958e39b68c03ae3b39a12400bbfe208b71d-300x272.jpg 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175636\/577e8958e39b68c03ae3b39a12400bbfe208b71d-65x59.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175636\/577e8958e39b68c03ae3b39a12400bbfe208b71d-225x204.jpg 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/10\/30175636\/577e8958e39b68c03ae3b39a12400bbfe208b71d-350x317.jpg 350w\" sizes=\"(max-width: 487px) 100vw, 487px\" \/><\/li>\n<\/ol>\n<p>Real-World Applications<\/p>\n<ol start=\"26\">\n<li>The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as [latex]x^2 = 4y[\/latex]. At what coordinates should you place the light bulb?<\/li>\n<li>A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is [latex]12[\/latex] feet across at its opening and [latex]4[\/latex] feet deep at its center, where should the receiver be placed?<\/li>\n<li>The reflector in a searchlight is shaped like a paraboloid of revolution. A light source is located [latex]1[\/latex] foot from the base along the axis of symmetry. If the opening of the searchlight is [latex]3[\/latex] feet across, find the depth.<\/li>\n<li>An arch is in the shape of a parabola. It has a span of [latex]100[\/latex] feet and a maximum height of [latex]20[\/latex] feet. Find the equation of the parabola, and determine the height of the arch [latex]40[\/latex] feet from the center.<\/li>\n<li>An object is projected so as to follow a parabolic path given by [latex]y = -x^2 + 96x[\/latex], where [latex]x[\/latex] is the horizontal distance traveled in feet and [latex]y[\/latex] is the height. Determine the maximum height the object reaches.<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":28,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":345,"module-header":"practice","content_attributions":[],"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\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5257"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5257\/revisions"}],"predecessor-version":[{"id":6003,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5257\/revisions\/6003"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/345"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/5257\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=5257"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=5257"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=5257"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=5257"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}