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8.5

8.5. Polar Equations of Conics DAY 1. Polar Coordinates - Review.

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8.5

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  1. 8.5 Polar Equations of Conics DAY 1

  2. Polar Coordinates - Review A polar coordinate systemis a plane with a point O, the , and a ray from O, the polar axis, as shown in Figure 6.35. Each point P in the plane is assigned as polar coordinates follows: r is the directed distancefrom O to P, and is the whose initial side is on the polar axis and whose terminal side is on the line OP.

  3. Plotting Points in the Polar Coordinate System

  4. Quick Review

  5. What you’ll learn about • Eccentricity Revisited • Writing Polar Equations for Conics • Analyzing Polar Equations of Conics • Orbits Revisited … and why You will learn the approach to conics used by astronomers.

  6. Focus-Directrix Definition Conic Section A conic section is the set of all points in a plane whose distances from a particular point (the focus) and a particular line (the directrix) in the plane have a constant ratio. (We assume that the focus does not lie on the directrix.)

  7. The line passing through the focus and perpendicular to the directrix is the (focal) axis of the conic section. The axis is a line of symmetry for the conic. The point where the conic intersects its axis is a vertex of the conic. If P is a point of the conic, F is the focus, and D is the point of the directrix closest to P, then the constant ratio PF/PDis the eccentricity, e, of the conic. A parabola has one focus and one directrix. Ellipses and hyperbolas have two focus-directrix pairs, and either focus-directrix pair can be used with the eccentricity to generate the entire conic section.

  8. The Geometric Structure of a Conic Section PF/PD is the eccentricity, e

  9. Focus-Directrix Eccentricity Relationship In this approach to conic sections, the eccentricity eis a strictly positive constant, and there are no circles or other degenerate conics.

  10. A Conic Section in the Polar Plane X = k let k be the focus to the directrix (remember, the focus is at the pole or center of the polar graph) k

  11. If the distance from the focus to the directrix is k, the Cartesian equation of the directrix is x = k. From Figure 8.41, we see that X = k k

  12. Three Types of Conics for r = ke/(1+ecosθ)

  13. Three Types of Conics for r = ke/(1+ecosθ)

  14. Three Types of Conics for r = ke/(1+ecosθ)

  15. Polar Equations for Conics The focal axis is x for cosine and y for sine. It also matches the equation of the directrix The sign of the denominator is the sign of the directrix. This give us the direction from the pole (0, 0)

  16. Polar Equations for Conics The focal axis is x for cosine and y for sine. It also matches the equation of the directrix The sign of the denominator is the sign of the directrix. This give us the direction from the pole (0, 0)

  17. Given that the focus is at the pole, write a polar equation for the specified conic, and graph it. (a) Eccentricity e=3/5, directrix x = 2.

  18. Eccentricity e=3/5, directrix x = 2.

  19. Graphing with the calculator

  20. Given that the focus is at the pole, write a polar equation for the specified conic, and graph it.

  21. Given that the focus is at the pole, write a polar equation for the specified conic, and graph it.

  22. Example Identifying Conics from Their Polar Equations

  23. Example Identifying Conics from Their Polar Equations

  24. Example Identifying Conics from Their Polar Equations

  25. Homework: Page 683: 1-19 all, 45, 47-50

  26. 8.5 Polar Equations of Conics DAY 2

  27. The cos tells us that the major or focal axis is the x axis. The vertices must be where the ellipse intersects the x axis. Since this is a polar equation, the vertices will be at 0 radians and p radians. Substitute 0 and p into the above to obtain the polar coordinates of the vertices.

  28. So the vertices are at (8,0) and (2,p)

  29. Keep in mind that all of the facts we learned about the conics in the previous sections still apply. Now that we have the vertices, we can find a. Recall that the length of the major axis of an ellipse is equal to 2a. To find the length of the major axis we need to find the distance between the vertices. In a Cartesian system we would subtract the coordinates. Since the vertices are polar, the sign of r indicates the direction. (r, q ) is r units in the direction of q. . (-r, q ) is r units in the direction of -q. In this problem we would ADD the coordinates. For (8, 0) and (2,p), we add the 8 and 2. The length of the major axis is 10. Since 2a = 10, a = 5. .

  30. As we can see from the previous slide, the distance from the focus to the vertex is equal to a – c. Since the focus is at the pole, and the nearest vertex is (2,p), we know the distance from the focus to the vertex is 2. Therefore a – c = 2; 5 – c = 2, so c = 3.

  31. Elliptical Orbits Around the Sun

  32. Semimajor Axes and Eccentricities of the Planets 1 gigameter(Gm) = 1 billion meters = 621,371.193 miles

  33. Ellipse with Eccentricity e and Semimajor Axis a To use the data in Table 8.3 to create polar equations for the elliptical orbits of the planets, we need to express the equation r = ke/(1+e cos q)in terms of a and e.We apply the formula PF e • PD to the ellipse shown in Figure 8.45:

  34. Ellipse with Eccentricity e and Semimajor Axis a

  35. A = 57.9 and e = .2056

  36. The orbit of the planet Uranus has a semimajor axis of 19.19 AU and an orbital eccentricity of 0.0461. Compute its perihelion and aphelion distances. 1 Astronomical Unit(AU) is defined as the mean distance between Earth and the sun or 149,597,870,700 meters or 92,955,807.3 miles

  37. The orbit of the planet Uranus has a semimajor axis of 19.19 AU and an orbital eccentricity of 0.0461. Compute its perihelion and aphelion distances.

  38. Halleys Comet Halley's Comet is arguably the most famous comet. It is a "periodic" comet and returns to Earth's vicinity about every 75 years, making it possible for a human to see it twice in his or her lifetime. The last time it was here was in 1986, and it is projected to return in 2061. The comet is named after English astronomer Edmond Halley, who examined reports of a comet approaching Earth in 1531, 1607 and 1682. He concluded that these three comets were actually the same comet returning over and over again, and predicted the comet would come again in 1758. Halley didn't live to see the comet's return, but his discovery led to the comet being named after him. (The traditional pronunciation of the name usually rhymes with valley.) Halley's calculations showed that at least some comets orbit the sun. Further, the first Halley's Comet of the space age – in 1986 – saw several spacecraft approach its vicinity to sample its composition. High-powered telescopes also observed the telescope as it swung by Earth.

  39. Halleys Comet Halley’s comet has an elliptical orbit with an eccentricity of e≈ 0.967. The length of the major axis of the orbit is approximately 35.88 astronomical units. Find a polar equation for the orbit. How close does Halley’s comet come to the sun? The next return of Halley’s comet will be in 2061. How old will you be?

  40. Homework: Page 683: 21-37 odd

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