If you add these two distances together, you always get the same number. It doesn’t matter which point on the edge you pick, as long as it is on the outer edge. (You’ll see in a bit why one of the focal points above is a star.) 28. For the ellipse shown above, measure the two distances and to the focal points pointed to by arrows on the dashed lines, and add them together. Then pick another arbitrary point on the perimeter of the ellipse, and repeat the same procedure. Do you always get the same sum? Handout 2 • p . 5 Lesson 5 (MATHEMATICS) Problems on Conic Sections 29. Figure out a way to draw an ellipse with a piece of cardboard, two thumbtacks, a loop of string, and a pen or pencil. 30. Pick a random point on the ellipse above problem #28. Using a ruler, verify that . (Consult question #23 for a reminder of what “a” is.) Is this true in general? Support your choice with a diagram. 31. What happens to the focal points of an ellipse as you squeeze an ellipse back to a circle? What happens to the big radius and the small radius? Johannes Kepler published work in the early 17th century that summarized three observational facts about the motion of the planets. These three facts have since been given the name “Kepler’s laws of planetary motion.” The first of these laws is that the orbit of a planet traces out an ellipse, where the sun is located at one of the focal points of the ellipse. (The other focal point doesn’t appear to play any role in an orbit. It’s almost always just an ordinary, unassuming location in empty space.)
Journeys in Film : Hidden Figures 32. Perhaps you’ve heard of black holes. If not, go look them up right now—they’re fascinating objects. The region around many black holes is often actually quite bright, because matter falling into it is heated to extremely high temperatures. However, if there is no nearby matter, a black hole is truly black, and is sometimes referred to as “dormant,” or “quiescent.” If you were able to accurately observe the path of a planet orbiting a dormant black hole in a highly elliptical orbit, would you be able to tell at which focus the black hole was located? The distance between each focal point and the center of an ellipse doesn’t have a commonly used name. Nevertheless, it’s a useful quantity. In the ellipse here, this distance is . The closest distance a planet gets to the sun is called the “perihelion,” and the farthest distance is called the “aphelion.” (The ancient Greek word for the sun was helios .) In the ellipse below (identical to the one above), the perihelion is labeled , and the aphelion is labeled . (Don’t confuse it with “a,” the big radius of the ellipse.) Handout 2 • p . 6 Lesson 5 (MATHEMATICS) Problems on Conic Sections 33. Using a ruler, verify that for this ellipse. Is this true in general? Support your choice with a diagram.
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