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 6.1 Circles and Parabolas • Conic Sections – Parabolas, circles, ellipses, hyperbolas 1 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 lengthr = (x − h)2 + (y − k)2 to some point (x, y) on the circle. • Squaring both sides yields the center- radius form of the equation of a circle. r 2 = (x − h)2 + ( y − k)2 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. 2 Solution Substitute h = –3, k = 4, and r = 6 into the equation of a circle. 62 = (x − (−3))2 + (y − 4)2 36 = (x + 3)2 + (y − 4)2 6.1 Graphing Circles with the Graphing Calculator Example Use the graphing calculator to graph the circle in a square viewing window. x2 + y2 = 9 3 Solution x2 + y2 = 9 y2 = 9 − x2 y = ± 9 − x2 2 2 Let y1 = 9 − x and y2 = − 9 − x . 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. 4 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. 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. 5 6.1 Determining Information about Parabolas from Equations Example Find the focus, directrix, vertex, and axis of each parabola. (a) x2 = 8y (b) y2 = −28x Solution (a) Since the x-term is squared, the 4c = 8 parabola is vertical, with focus c = 2 at (0, c) = (0, 2) and directrix y = –2. The vertex is (0, 0), and the axis is the y-axis. 6 6.1 Determining Information about Parabolas from Equations (b) 4c = −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. 6.1 Writing Equations of Parabolas Example Write an equation for the parabola with vertex (1, 3) and focus (–1, 3). 7 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)2 = 4(−2)(x −1) (y − 3)2 = −8(x −1) 8.