Answered: Focal chords A focal chord of a conic…

College Algebra

7th Edition

ISBN:9781305115545

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter7: Conic Sections

Section7.4: Shifted Conics

Problem 68E: DISCUS: A Family of Confocal Conics Conics that share a focus are called confocal. Consider the...

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Reflecting Telescope The Hale telescope at the Mount Palomar Observatory has a 200-in. mirror, as shown in the figure. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.
Tour Boat A tour boat travels between two islands that are 12 miles apart (see figure). For each trip between the islands, there is enough fuel for a 20-mile trip. (a) Explain why the region in which the boat can travel is bounded by an ellipse. (b) Let (0,0) represent the center of the ellipse. Find the coordinates of the center of each island. (c) The boat travels from Island 1, past Island 2 to one vertex of the ellipse, and then to Island 2 (see figure). How many miles does the boat travel? Use your answer to find the coordinates of the vertex. (d) Use the results of parts (b) and (c) to write an equation of the ellipse that bounds the region in which the boat can travel.
DISCUS: A Family of Confocal Conics Conics that share a focus are called confocal. Consider the family of conics that have a focus at (0, 1) and a vertex at the origin, as shown in the figure. Find equations of two different ellipses that have these properties. Find equations of two different hyperbolas that have these properties. Explain why only one parabola satisfies these properties. Find its equation. Sketch the conics you found in parts (a), (b), and (c) on the same coordinate axes (for the hyperbolas, sketch the top branches only). How are the ellipses and hyperbolas related to the parabola?

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Focal chords A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. The length of the latus rectum of the parabola y2 = 4px or x2 = 4py is 4 | p | .