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Draw an ellipse with a shoe string and two points.

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Presentation Transcript
slide2

The shape of a spotlight on a planar surface is in most cases an ellipse. In some cases it may be a circle. If you cut a cylinder at an angle, you will get elliptical sections. This can have important applications in optics (lenses and mirrors can be elliptical in shape), or in the kitchen (where one might cut vegetables or sausage along a "bias cut" in order to obtain pieces that have the same thickness, but have more surface area exposed. Some tanks are in fact elliptical (not circular) in cross section. This gives them a high capacity, but with a lower center-of-gravity, so that they are more stable when being transported. And they're shorter, so that they can pass under a low bridge. You might see these tanks transporting heating oil or gasoline on the highway The ellipse is found in art and architecture, and you may be familiar with the Ellipse, part of a President's Park South (a National Park in Washington, DC, just south of the White House). Ellipses (or half-ellipses) are sometimes used as fins, or airfoils in structures that move through the air. The elliptical shape reduces drag. On a bicycle, you might find a chainwheel (the gear that is connected to the pedal cranks) that is approximately elliptical in shape. Here the difference between the major and minor axes of the ellipse is used to account for differences in the speed and force applied, because your legs push and pull more effectively when the pedals are arranged so that one pedal is in front and one is in back, than when the pedals are in the "dead zone" (when one pedal is up and one pedal is down).

10 4 ellipses

10.4 Ellipses

Purpose

Graph and write equations of ellipse.

Find the center, foci, major axis.

slide5

An ellipse is the set of all points P such that the sum of the distances between P and two distinct fixed points, called the foci is a constant

slide6
The line through the foci intersects the ellipse at two points, the vertices. The line through the vertices is the major axis.
slide7

The line perpendicular to the major axis at the center intersects the ellipse at two points called the co-vertices. The line segment joining these points is the minor axis.

write an equation of the ellipse with the given characteristics and center at 0 0
Write an equation of the ellipse with the given characteristics and center at (0,0)
  • Vertex (0,7) Co-vertex (-6,0)
write an equation of the ellipse with the given characteristics and center at 0 012
Write an equation of the ellipse with the given characteristics and center at (0,0)
  • Vertex (-4,0) Co-vertex (2,0)
write an equation of the ellipse with the given characteristics and center at 0 013
Write an equation of the ellipse with the given characteristics and center at (0,0)
  • Vertex (0,7) Foci (0,3)
write an equation of the ellipse with the given characteristics and center at 0 014
Write an equation of the ellipse with the given characteristics and center at (0,0)
  • Co-vertex ( ,0) Focus: (0,-3)
slide16
A portion of the White House lawn is called the Ellipse. It is 1060feet ling and 890 feet wide.
  • Write an equation of The Ellipse.
  • The area of an ellipse is A=πab. What is the area of the Ellipse at the White House?
slide17
In its elliptical orbit, Mercury ranges from 46.04 million kilometers to 69.86 million kilometers from the center of the sun. The center of the sun is a focus of the orbit. Write an equation of the orbit.