Conics Ellipses. Ellipse Definition: the set of all points in a plane whose distances from two fixed points in the plane have a constant sum. The fixed.

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Presentation transcript:

Conics Ellipses

Ellipse Definition: the set of all points in a plane whose distances from two fixed points in the plane have a constant sum. The fixed points are the foci of the ellipse. The line through the foci is called the focal axis. The point on the focal axis are the vertices of the ellipse.

To find c use the Pythagorean relation: a²=b²+c²

Ellipse with center (0,0) HorizontalVertical Focal Axisx-axis y=0 y-axis x=0 Foci (±c,0)(0, ±c) Vertices (±a,0)(0, ±a) Semi major axis a a Semi minor axis b b To find c use the Pythagorean relation: a²=b²+c²

E1 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

E1

E2 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci 64x² + 4y² = 256

E2

Ellipse with center (h,k)

Ellipse with center (0,0) HorizontalVertical Focal Axis y=k x=h Foci (h±c,k)(h, k±c) Vertices (h±a,k)(h, k±a) Semi major axis a a Semi minor axis b b To find c use the Pythagorean relation: a²=b²+c²

E3 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

E3

E4 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

E4

E5 Graph the ellipse. Label the center and 4 other points. Then list the vertices and foci

E5

Eccentricity The eccentricity is the ratio of c to a e=c/a