3. Show that C is a hyperbola if e > 1. 4. Show that the polar equation ed 1 + e cos 0 represents an ellipse if e 1. 5. For each of the following conics, find the eccentricity and directrix. Then identify and sketch the conic. 4 8 2 (a) r = 1 + 3 cos 0 ( b) r = 3 + 3 cos 0 (c) r = 2 + cos 0 6. Graph the conics r = e/(1 - e cos 0) with e = 0.4, 0.6, 0.8, and 1.0 on a common screen. How does the value of e affect the shape of the curve? 7. (a) Show that the polar equation of an ellipse with directrix x = d can be written in the form a(1 - ez) 1 - e cos 0 (b) Find an approximate polar equation for the elliptical orbit of the planet Earth around the sun (at one focus) given that the eccentricity is about 0.017 and the length of the major axis is about 2.99 X 108 km. planet 8. (a) The planets move around the sun in elliptical orbits with the sun at one focus. The positions of a planet that are closest to and farthest from the sun are called its peri- helion and aphelion, respectively. (See Figure 2.) Use Problem 7(a) to show that the perihelion distance from a planet to the sun is a(1 - e) and the aphelion distance sun is a(1 + e). aphelion perihelion (b) Use the data of Problem 7(b) to find the distances from the planet Earth to the sun at perihelion and at aphelion. 9. (a) The planet Mercury travels in an elliptical orbit with eccentricity 0.206. Its minimum FIGURE 2 distance from the sun is 4.6 X 10' km. Use the results of Problem 8(a) to find its maxi- mum distance from the sun. (b) Find the distance traveled by the planet Mercury during one complete orbit around the sun. (Use your calculator or computer algebra system to evaluate the definite integral.)DISCOVERY PROJECT Conic Sections in Polar Coordinates In this project we give a unified treatment of all three types of conic sections in terms of a focus and directrix. We will see that if we place the focus at the origin, then a conic section has a simple polar equation. In Chapter 10 we will use the polar equation of an ellipse to derive Kepler's laws of planetary motion. Let F be a fixed point (called the focus) and I be a fixed line (called the directrix) in a plane. Let e be a fixed positive number (called the eccentricity). Let C be the set of all points P in the I (directrix) plane such that P PF | PI x = d A (that is, the ratio of the distance from F to the distance from / is the constant e). Notice that if F the eccentricity is e = 1, then | PF | = [ P/ | and so the given condition simply becomes the 7 COS definition of a parabola as given in Appendix B. 1. If we place the focus F at the origin and the directrix parallel to the y-axis and d units to the right, then the directrix has equation x = d and is perpendicular to the polar axis. If the point P has polar coordinates (r, 0), use Figure 1 to show that 1 = eld - r cos 0) 2. By converting the polar equation in Problem 1 to rectangular coordinates, show that the FIGURE 1 curve C is an ellipse if e