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Chapter 6: Analytic Geometry 6.1 Circles and Parabolas 6.2 Ellipses and Hyperbolas 6.3 Summary of the Conic Sections 6.4 Parametric Equations Copyright © 2007 Pearson Education, Inc. Slide 6-2 6.1 Circles and Parabolas • Conic Sections – Parabolas, circles, ellipses, hyperbolas Copyright © 2007 Pearson Education, Inc. Slide 6-3 6.1 Circles A circle is a set of points in a plane that are equidistant from a fixed point. The distance is called the radius of the circle, and the fixed point is called the center. • • A circle with center (h, k) and radius r has length 2 2 r ( x h ) ( y k ) to some point (x, y) on the circle. Squaring both sides yields the center-radius form of the equation of a circle. r ( x h) ( y k ) 2 Copyright © 2007 Pearson Education, Inc. 2 2 Slide 6-4 6.1 Finding the Equation of a Circle Example Find the center-radius form of the equation of a circle with radius 6 and center (–3, 4). Graph the circle and give the domain and range of the relation. Solution Substitute h = –3, k = 4, and r = 6 into the equation of a circle. 6 ( x ( 3 )) ( y 4 ) 2 2 36 ( x 3 ) ( y 4 ) 2 Copyright © 2007 Pearson Education, Inc. 2 2 Slide 6-5 6.1 Graphing Circles with the Graphing Calculator Example Use the graphing calculator to graph the circle x 2 y 2 9 in a square viewing window. Solution x y 9 2 2 y 9x 2 2 y 9x Let y1 Copyright © 2007 Pearson Education, Inc. 2 9 x and y 2 9 x . 2 2 Slide 6-6 6.1 Graphing Circles with the Graphing Calculator • TECHNOLOGY NOTES: – Graphs in a nondecimalwindow may not be connected – Graphs in a rectangular (non-square) window look like an ellipse Copyright © 2007 Pearson Education, Inc. Slide 6-7 6.1 Finding the Center and Radius of a Circle Example Find the center and radius of the circle with equation x 2 6 x y 2 10 y 25 0. Solution Our goal is to obtain an equivalent equation of the form r 2 ( x h ) 2 ( y k ) 2 . We complete the square in both x and y. x 6 x y 10 y 25 2 2 ( x 6 x 9) ( y 10 y 25) 25 9 25 2 2 ( x 3) ( y 5) 9 2 2 ( x 3) ( y 2) 3 2 2 2 The circle has center (3, –2) with radius 3. Copyright © 2007 Pearson Education, Inc. Slide 6-8 6.1 Equations and Graphs of Parabolas A parabola is a set of points in a plane equidistant from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line, the directrix. • For example, let the directrix be the line y = –c and the focus be the point F with coordinates (0, c). Copyright © 2007 Pearson Education, Inc. Slide 6-9 6.1 Equations and Graphs of Parabolas • To get the equation of the set of points that are the same distance from the line y = –c and the point (0, c), choose a point P(x, y) on the parabola. The distance from the focus, F, to P, and the point on the directrix, D, to P, must have the same length. d (P, F ) d (P, D) ( x 0) ( y c) ( x x ) ( y ( c )) x y 2 yc c y 2 yc c 2 2 2 2 2 2 2 x y 2 yc c y 2 yc c 2 2 2 2 2 2 2 x 4 cy 2 Copyright © 2007 Pearson Education, Inc. Slide 6-10 6.1 Parabola with a Vertical Axis The parabola with focus (0, c) and directrix y = –c has equation x2 = 4cy. The parabola has vertical axis x = 0, opens upward if c > 0, and opens downward if c < 0. • The focal chord through the focus and perpendicular to the axis of symmetry of a parabola has length |4c|. – Let y = c and solve for x. x 4 cy 2 x 4c 2 2 x 2 c or 2 c The endpoints of the chord are ( x, c), so the length is |4c|. Copyright © 2007 Pearson Education, Inc. Slide 6-11 6.1 Parabola with a Horizontal Axis The parabola with focus (c, 0) and directrix x = –c has equation y2 = 4cx. The parabola has horizontal axis y = 0, opens to the right if c > 0, and to the left if c < 0. • Note: a parabola with a horizontal axis is not a function. • The graph can be obtained using a graphing calculator by solving y2 = 4cx for y: y 2 cx . Let y1 2 cx and y 2 2 cx and graph each half of the parabola. Copyright © 2007 Pearson Education, Inc. Slide 6-12 6.1 Determining Information about Parabolas from Equations Example Find the focus, directrix, vertex, and axis of each parabola. 2 (a) x 2 8 y (b) y 28 x Solution (a) 4 c 8 c 2 Since the x-term is squared, the parabola is vertical, with focus at (0, c) = (0, 2) and directrix y = –2. The vertex is (0, 0), and the axis is the y-axis. Copyright © 2007 Pearson Education, Inc. Slide 6-13 6.1 Determining Information about Parabolas from Equations (b) 4 c 28 c 7 The parabola is horizontal, with focus (–7, 0), directrix x = 7, vertex (0, 0), and x-axis as axis of the parabola. Since c is negative, the graph opens to the left. Copyright © 2007 Pearson Education, Inc. Slide 6-14 6.1 Writing Equations of Parabolas Example Write an equation for the parabola with vertex (1, 3) and focus (–1, 3). Solution Focus lies left of the vertex implies the parabola has - a horizontal axis, and - opens to the left. Distance between vertex and focus is 1–(–1) = 2, so c = –2. ( y 3 ) 4 ( 2 )( x 1) 2 ( y 3 ) 8 ( x 1) 2 Copyright © 2007 Pearson Education, Inc. Slide 6-15 6.1 An Application of Parabolas Example Signals coming in parallel to the axis of a parabolic reflector are reflected to the focus, thus concentrating the signal. The Parkes radio telescope has a parabolic dish shape with diameter 210 feet and depth 32 feet. Copyright © 2007 Pearson Education, Inc. Slide 6-16 6.1 An Application of Parabolas (a) Determine the equation describing the cross section. (b) The receiver must be placed at the focus of the parabola. How far from the vertex of the parabolic dish should the receiver be placed? Solution (a) The parabola will have the form y = ax2 (vertex at the origin) and pass through the point 2102 , 32 (105 , 32 ). 32 a (105 ) a 32 105 2 32 32 y 2 , so the cross section can be described by 11 , 025 2 x . 11 , 025 Copyright © 2007 Pearson Education, Inc. Slide 6-17 6.1 An Application of Parabolas (b) Since y 32 2 x , 11 , 025 4c 1 a 4c 11 , 025 32 c 11 , 025 86 . 1 . 128 The receiver should be placed at (0, 86.1), or 86.1 feet above the vertex. Copyright © 2007 Pearson Education, Inc. Slide 6-18