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Rev.S08 MAC 1140 Module 11 Conic Sections

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2 Rev.S08 Learning Objectives Upon completing this module, you should be able to find equations of parabolas. graph parabolas. use the reflective property of parabolas. translate parabolas. find equations of ellipses. graph ellipses. use the reflective property of ellipses. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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3 Rev.S08 Learning Objectives (Cont.) 8. find the center and radius of a circle. 9. solve systems of nonlinear equations and inequalities. 10. find equations of hyperbolas. 11. graph hyperbolas. 12. use the reflective property of hyperbolas. 13. translate hyperbolas. 14. solve systems of nonlinear equations. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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4 Rev.S08 Conic Sections http://faculty.valenciacc.edu/ashaw/ http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. 10.1Parabolas 10.2Ellipses 10.3 Hyperbolas There are three sections in this module:

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5 Rev.S08 What are Conic Sections? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Conic Sections are named after the different ways a plane can intersect a cone. The three basic conic sections are parabolas, ellipses, and hyperbolas. A circle is an example of an ellipse.

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6 Rev.S08 What are Focus and Directrix? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. In previous module, we learned that a parabola with vertex (0,0) can be represented symbolically by y = a x 2. A parabola is the set of points in a plane that are equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola. In this graph, the parabolas has a vertical axis of symmetry; the fixed point is (0, -1) and the fixed line is y = 1. y = 1 Focus

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7 Rev.S08 Equation of a Parabola with Vertex (0,0) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. If the value of p is known, then the equation of a parabola with vertex (0,0) can be written as one of the following equations: Vertical axis of symmetry: The parabola with a focus at (0,p) and directrix y = -p has equation x 2 = 4py. The parabola opens upward if p > 0 and downward if p < 0. Horizontal axis of symmetry: The parabola with a focus at (p,0) and directrix x = -p has equation y 2 = 4px. This parabola opens to the right if p > 0 and to the left if p < 0.

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8 Rev.S08 Example of Finding Focus and Directrix http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let’s try to find the focus and directrix for x 2 = -4y. Since this parabola has a vertical axis of symmetry, it has equation x 2 = 4py so, −4 = 4p −1 = p Since p < 0, the parabola opens downward, and Focus: (0, p) = (0, −1) Directrix: y = - p y = -(-1)=1 y = 1 Focus

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9 Rev.S08 Example of Finding Focus and Directrix http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Let’s try to find the focus and directrix for y 2 = 8x. Since this parabola has a horizontal axis of symmetry, it has equation y 2 = 4px so, 8 = 4p = p Since p > 0, the parabola opens to the right, and Focus: (p, 0) = (2, ) Directrix: x = - p x = - 2 x = −2 F(2, 0) V(0, 0)

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10 Rev.S08 How to Find the Equation of the Parabola? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Find the equation of the parabola with focus (0.5, 0) and vertex at the origin. Solution The given focus is at (p, 0), so the parabola has a horizontal axis of symmetry. The parabola opens to the right since p = 0.5 (p > 0.) The equation of this parabola: y 2 = 4px y 2 = 4(0.5)x y 2 = 2x

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11 Rev.S08 Reflective Property of Parabola http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. When a parabola is rotated about its axis, it sweeps out a shape called a paraboloid. Paraboloids have a special reflective property. When incoming parallel rays of light from the sun or distant star strike the surface of a paraboloid, each ray is reflected toward the focus.

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12 Rev.S08 How to Use Translations of Graphs to Find the Equation of a Parabola? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. If the equation of a parabola is given by x 2 = 4py or y 2 = 4px, we know the vertex is (0,0). If the vertex is (h,k), we can use translations of graphs to find the equation of the parabola. This equation can be obtained by replacing x with (x - h) and replacing y with (y - k).

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13 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Graph the parabola Solution Rewrite the equation in standard form. The equation of this parabola: (y - k) 2 = 4p(x - h) Vertex: (h, k) = (1, 3) 4p = 12 p > 0 opens right p = 3 Focus: (h+p,k) = (1+3,3) = (4, 3) Directrix: x = h - p = 1 - 3 = −2

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14 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Find the equation of the parabola with focus (−2, 3) and directrix y = −3. Solution Sketch the focus and directrix. Parabola opens up with vertical axis of symmetry. Focus: (h, k + p) = (-2, 3), so h = -2, k + p= 3 (1) Directrix: y = k - p = -3 (2) Add Equation (1) to Equation (2) => k = 0, p = 3 Vertex: (h, k) = (−2, 0) Distance from focus to vertex is 3. Equation: (x - h) 2 = 4py => (x + 2) 2 = 12y

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15 Rev.S08 How to Find the Equation of a Parabola by Completing the Square? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Write x 2 + 6x + 4y + 5 = 0 in the form (x − h) 2 = a(y − k). Solution Complete the square

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16 Rev.S08 What is an Ellipse? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. An ellipse is the set of points in a plane, the sum of whose distances from two fixed points is constant. Each fixed point is called a focus (plural foci) of the ellipse. The major axis is the longer axis, it can be either horizontal or vertical.

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17 Rev.S08 What are the Standard Equations for Ellipses Centered at the Origin? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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18 Rev.S08 How to Graph Ellipses? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Horizontal Major AxisVertical Major Axis

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19 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Sketch the graph of the ellipse 9x 2 + 4y 2 = 36. Solution Foci: Endpoints of minor axis ( 2, 0) Vertical major axis a = 2 and b = 3 c 2 = 3 2 − 2 2 = 5 or c = Vertices: (0, 3) foci

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20 Rev.S08 Example of Finding the Standard Equation of an Ellipse http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Find the standard equation of the ellipse with foci (0, 3) and major axis length 8. Solution 2a = 8 a = 4 c = 3 b 2 = a 2 – c 2 = 4 2 – 3 2 = 7 Focus on y-axis use larger intercept for denominator of y 2.

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21 Rev.S08 Reflective Property of Ellipses http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Like parabolas, ellipses also have an important reflective property. If an ellipse is rotated about the x-axis, an ellipsoid is formed, which resembles the shell of an egg. If a light source is placed at focus F 1, then every beam of light emanating from the light source, regardless of direction, is reflected at the surface of an ellipsoid toward focus F 2.

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22 Rev.S08 What are the Standard Equations for Ellipses? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. As we can see here, we use translations of graphs to write the equation of ellipses centered at (h,k) by replacing x with (x - h) and replacing y with (y - k) from the standard equations for ellipses centered at the origin.

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23 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Graph the ellipse whose equation is Solution Vertical major axis Center: (2, −1) a 2 = 16, b 2 = 9, c 2 = 7 Vertices are 4 units above and below the center. (2, 3)(2, −5). Foci: center (2, −1)

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24 Rev.S08 Example of Using Completing the Square to find the Standard Form for an Ellipse http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Write 4x 2 + 9y 2 − 16x + 18y − 11 = 0 in the standard form for an ellipse centered at (h, k). Identify the center and vertices. Solution Complete the square: Center (2, −1) Vertices: (−1, −1) and (5, −1)

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25 Rev.S08 What is the Standard Form of a Circle? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The standard form of a circle with center (h, k) and radius r is (x – h) 2 + (y – k) 2 = r 2. Example: Find the center and radius of a circle given by x 2 + y 2 − 14x + 4y = 11. Solution center: (7, −2) radius: 8

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26 Rev.S08 How to Find the Area Inside an Ellipse? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Given the standard equation of an ellipse, the area A of the region contained inside is given by A = ab. Example: Shade the region in the xy-plane that satisfies the inequality 9x 2 + 4y 2 < 36. Find the area of this region if units are in inches. Solution Write in standard form:

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27 Rev.S08 How to Find the Area Inside an Ellipse? (cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The region inside the ellipse satisfies the inequality. A = ab A = (2)(3) 18.85 square inches

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28 Rev.S08 What is a Hyperbola? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points is constant. Each fixed point is called a focus of the hyperbola. The transverse axis is the line segment connecting the vertices, and its length equals 2a. There are two branches and asymptotes to a hyperbola.

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29 Rev.S08 What is a Hyperbola? (cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The top hyperbola is having a horizontal transverse axis, and its length equals 2a. There are one left branch and one right branch. The bottom hyperbola is having a vertical transverse axis, and its length equals 2a. There are one upper branch and one lower branch. A hyperbola consists of two solid curves or branches. The other parts are aids for sketching its graph. Note: The dashed diagonal axes are asymptotes. The dashed rectangle is sometimes called the fundamental rectangle.

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30 Rev.S08 What are the Standard Equations for Hyperbolas centered at the Origin? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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31 Rev.S08 How to Graph a Hyperbola? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Sketch the graph of Solution a = 5 and b = 7 c 2 = a 2 + b 2 c 2 = 25 + 49 = 74 Asymptotes: Vertices: ( 5, 0) Foci:

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32 Rev.S08 How to Find the Equation of the Hyperbola centered at the Origin? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Find the equation of the hyperbola centered at the origin with vertices (0, 4) and foci (0, 6). Solution a = 4, c = 6 b 2 = c 2 – a 2 = 6 2 – 4 2 = 20

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33 Rev.S08 Reflective Property of Hyperbolas http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Hyperbolas have an important reflective property. If a hyperbola is rotated about the x-axis, a hyperboloid is formed. Any beam of light that is directed toward focus F 1, will be reflected by the hyperboloid toward focus F 2.

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34 Rev.S08 What are the Standard Equations for Hyperbolas Centered at (h, k)? http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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35 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Graph the hyperbola Solution a 2 = 9, b 2 = 4 c 2 = a 2 + b 2 c 2 = 9 + 4 = 13 Center: (−3, −2) Vertices: (−3, −2 3) Foci: Asymptotes:

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36 Rev.S08 Example http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Write 4x 2 + 8x – y 2 – 4y = 4 in standard form for a hyperbola centered at (h, k). Identify the center and the vertices. Solution Center: (−1, −2) Vertices:

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37 Rev.S08 What have we learned? We have learned to find equations of parabolas. graph parabolas. use the reflective property of parabolas. translate parabolas. find equations of ellipses. graph ellipses. use the reflective property of ellipses. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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38 Rev.S08 What have we learned? (Cont.) 8. find the center and radius of a circle. 9. solve systems of nonlinear equations and inequalities. 10. find equations of hyperbolas. 11. graph hyperbolas. 12. use the reflective property of hyperbolas. 13. translate hyperbolas. 14. solve systems of nonlinear equations. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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39 Rev.S08 Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Rockswold, Gary, Precalculus with Modeling and Visualization, 3th Edition http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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