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Notes 8.2 Conics Sections – The Ellipse

Notes 8.2 Conics Sections – The Ellipse. I. Introduction. A.) Def: The set of all points in a plane whose distances from two fixed points in the plane have a constant sum. 1.) The fixed points are the FOCI . 2.) The line through the foci is the FOCAL AXIS.

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Notes 8.2 Conics Sections – The Ellipse

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  1. Notes 8.2 Conics Sections – The Ellipse

  2. I. Introduction A.) Def: The set of all points in a plane whose distances from two fixed points in the plane have a constant sum. 1.) The fixed points are the FOCI. 2.) The line through the foci is the FOCAL AXIS. 3.) The CENTER is ½ way between the foci and/or the vertices.

  3. B.) Forming an Ellipse - When a plane intersects a double-napped cone and is neither parallel nor perpendicular to the base of the cone, an ellipse is formed.

  4. P(x, y) Focus Focus C.) Pictures – By Definition - (x, y) d1 d2 (F1, 0) (F2, 0)

  5. Minor Axis Center Focus Pictures -Expanded- (0, b) Focus Major Axis Vertex Vertex (-a, 0) (-c, 0) (0, 0) (c, 0) (a, 0) (0, -b) b is the SEMI- MINOR axis a is the SEMI- MAJOR axis

  6. D.) Standard Form Equation - Where b2+ c2 = a2.

  7. E.) ELLIPSES - Center at (0,0) St. Fm. Focal axis Foci Semi-Major Semi-Minor Pyth. Rel.

  8. F.) ELLIPSES - Center at (h, k) St. fm. Focal axis Foci Semi-Major Semi-Minor Pyth. Rel.

  9. II. Examples A. ) Ex. 1- Find the vertices and foci of the following ellipse: Vertices = Foci =

  10. B.) Ex. 2- Find a equation of the ellipse with foci (4,0) and (-4,0) whose minor axis has a length of 6.

  11. C.) Ex. 3- Find the center, foci, and vertices of the following ellipse:

  12. D.) Ex. 2- Find the equation of an ellipse with foci (-2, 1) and (-2, 5) and major-axis endpoints (-2, -1) and (-2,7).

  13. III. Eccentricity A.) B.) What it tells us – 1.) e close to 0  foci close to center 2.) e close to 1  foci close to vertices

  14. IV. Ellipsoids of Revolution A.) Rotate ellipse about its focal axis to get an ellipsoid of revolution B.) Examples of these include whispering galleries and a lithotripter, a device which uses shockwaves to destroy kidney stones.

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