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Chapter 11: Introduction When a right circular cone is cut by a plane, the intersection of the two is called a conic section. (If the figure formed is a point, line, or intersecting lines, we call this a degenerate conic section.) Conic sections are used to model many natural occurring situations such as planet orbits. Section 11.1: Ellipses An ellipse can be thought of as a circle that has been stretched horizontally or vertically. All ellipses have two foci (singular: focus). The sum of the distances from the foci to any point on the ellipse is a constant. The center of an ellipse is located directly between the two foci. An ellipse has two axes. The major axis passes through the foci and the center and connects the outermost points of the ellipse which are called vertices. The minor axis is perpendicular to the major axis through the center and connects to the co-vertices. Below is the equation and characteristics of an ellipse that has a center at the origin: EX1: Graph and label the parts of the following ellipse 25 x 2 16 y 2 400 EX2: Graph the following ellipse on a calculator 4 x 2 9 y 2 36 Note: To get an accurate shape for your conic graphs use zoom square EX3: Find the equation of the ellipse with the following characteristics: a. Foci on the x-axis; x-intercepts ±7; y-intercepts ±2 b. Foci on the y-axis; major axis of length 20; minor axis of length 18 EX4: Halley’s Comet has an elliptical orbit with the sun at one focus and a major axis of 1,636,484,848 miles. The closest the comet comes to the sun is 54,004,000 miles. What is the maximum distance from the comet to the sun? Section 11.2: Hyperbolas A hyperbola, like the ellipse, has two foci and the curve bends around these two points. The absolute value of the difference of the distance the foci to any point on the hyperbola is a constant. The hyperbola has two asymptotes which cross at the hyperbola’s center. The center is located half way in between the two foci. The hyperbola has an axis of symmetry that passes through the foci and the center. It is called the focal axis. The points where the focal axis passes through the hyperbola is called the vertices. Below is the equation and characteristics of a hyperbola that has a center at the origin: The a and b values can be used to create an auxiliary rectangle that will help graph the hyperbola. The asymptotes of the graph pass through the corners of the rectangle. EX1: Graph the following hyperbola: 9 x 2 16 y 2 144 EX2: Graph the following hyperbola on a calculator: 16 x 2 4 y 2 64 EX3: Find the equations of the following hyperbolas: a. x-intercepts: 3 , asymptote: y 2 x b. focus: (3,0) and vertex: (2,0) Section 11.3: Parabolas As we have learned in previous lessons, a parabola is the basic shape of a quadratic equation ( y ax 2 bx c ). In this lesson we will learn an alternate definition. A parabola is the set of all points equidistant from a point (the focus) and a line (the directrix). The extreme point of the parabola is called its vertex. A line that is perpendicular to the directrix passes through the vertex and the focus cuts the parabola in half. This line is the axis of symmetry. Below is the equation and characteristics of a parabola that has a center at the origin: EX1: Sketch and label the parts of 2 x 2 8 y EX2: Graph the parabola on a calculator: y 2 8 x EX3: Find the equation with the vertex at the origin with the following characteristics: a. axis: x 0 , passing through (3,12) b. focus: (0,8) Section 11.4: Translations and Rotations of Conics Horizontal and Vertical Shifts Adding or subtracting a constant to the x or y terms translates the conic Below are the standard equations of translated conics: EX1: Sketch the graph of the conic ( x 1) 2 ( y 5) 2 a. 1 4 9 b. y 4( x 1) 2 2 In order to identify a conic, write it in standard form by completing the square. EX2: Identify the conic and describe the graph a. 9 x 2 4 y 2 54 x 8 y 49 0 b. x 2 6 x y 5 0 EX3: Write the equation of the translated conic a. Ellipse with center (2,3) ; Endpoints of major and minor axis: (2, 1),(0,3),(2,7),(4,3) b. Hyperbola with center (4, 2) ; vertex: (7, 2) ; asymptote: 3 y 4 x 10 EX4: Graph the following conics on a graphing calculator: a. 4 y 2 x 2 6 x 24 y 11 0 b. 9 x 2 24 xy 16 y 2 90 x 130 y 0 Any second degree equation in x and y can be written in Standard Form: Ax 2 Bxy Cy 2 Dx Ey F 0 We can find out what type of conic we have by looking at the discriminant: B 2 4 AC Positive: Hyperbola (or Two Intersecting Lines if degenerate case) Negative: Circle, Ellipse (or Point if degenerate case) Zero: Parabola (or Line or Two Parallel Lines if degenerate case) EX5: Use the discriminant to determine the type of conic and graph to confirm your hypothesis: 23 x 2 26 3 xy 3 y 2 16 x 16 3 y 128 0 Section 11.5: Polar Coordinates Probably every graph that you have studied in the past has been on a Cartesian plane and the points looked like (x, y). In this lesson we will study a new type of graph called polar graphing. The points for polar graphing look like this: r , , where r stands for radius and theta is an angle. As seen below, the coordinate plane looks somewhat different as well. To plot polar coordinates we simply start at the origin and move down the polar axis (sometimes called the pole) till you get to the right radius. We then rotate the point around the origin by . EX1: Plot the following polar coordinates a. 2, 4 b. 3, 56 c. 1, 73 EX2: Determine if the given coordinates represent the same as 2, in the polar 3 coordinate system: a. 2, 73 b. 2, 23 c. 2, 43 To see how to change between polar and rectangular coordinates, we can put the two graphs on top of each other. Notice that the radius is simply the hypotenuse of a right triangle and by the Pythagorean theorem r x2 y 2 . Also, using trig we see that x y y cos sin tan r r x which leads to x r cos y r sin EX3: Convert the following polar coordinates from polar to rectangular. a. 3, 3 b. 1, 56 EX4: Convert the following rectangular coordinates to polar. a. 4,3 b. 3, 7 Polar graphing works the same way as Cartesian graphing when it comes to equations. The graphs of lines and circles are simpler in polar coordinates than in Cartesian. EX5: Graph the following polar equations a. r 2 b. c. r 1 2sin 3 Section 11.6: Polar Equations and Conics The eccentricity of a conic section describes its shape and how flat or spread out it is. It is defined by the following formula: distance between the foci c e distance between the vertices a The eccentricities of the different conics are listed below. Conic Eccentricity Circle 0 Ellipse 0 e 1 Parabola 1 Hyperbola e 1 EX1: Find the eccentricity of the conic whose equation is given. x2 y 2 a. 1 b. 16 x 2 32 x 9 y 2 36 y 124 0 100 99 Polar Equations of Conic Sections In order to define conics in terms of polar coordinates we must first define conics and eccentricity in a different way. We are going to let L be a line in a plane called a directrix. P is some fixed point that is not located on L. X is going to be the set of all points that makes up our shape. We can now define eccentricity in terms of these figures. distance between X and the fixed point XP e distance between X and the fixed line XL If d is the distance from the focus at the pole to the directrix, then the characteristics of polar conics are shown in the table. EX2: Identify the conic section from its polar form. If it is an ellipse or hyperbola, state its eccentricity. 12 6 a. r b. r 3 4sin 5 2cos 5 EX3: Sketch the graph of the conic: r 3 2sin EX4: Find the polar equation of the conic section that has a focus at (0,0) and satisfies the given condition: a. parabola; vertex 3, 3 b. ellipse; vertices 2, and 8, 2 2 c. hyperbola; vertices 1,0 and 3,