Ellipse.

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Presentation transcript:

ellipse

Ellipse The set of all co-planar points whose sum of the distances from two fixed points (foci) are constant.

Definition of major and minor axis Ellipse Definition of major and minor axis minor axis major axis

Ellipse major axis minor axis

Definition of vertices Ellipse Definition of vertices and co-vertices co-verticies verticies verticies endpoint of major axis co-verticies endpoint of minor axis

Ellipse a2 = b2 + c2 Distance from center to vertex = a Distance from center to co-vertex = b Distance from center to foci = c b c a Length of major axis = 2a Length of minor axis = 2b a2 = b2 + c2

Ellipse - horizontal a2 = b2 + c2 b c a Distance from center to vertex = a a2 = b2 + c2 Distance from center to co-vertex = b Distance from center to foci = c

Ellipse - vertical a2 = b2 + c2 a c b Distance from center to vertex = a Distance from center to co-vertex = b Distance from center to foci = c

Graph the following Ellipse Center: (2, -3) a = 5 in x direction b = 3 in y direction (2, 0) 3 (2, -3) (-3, -3) (7, -3) 5 5 3 (2, -6)

Graph the following Ellipse a = 5 b = 3 a2 = b2 + c2 52 = 32 + c2 25 = 9 + c2 foci 16 = c2 4 = c (2, 0) 3 (-2, -3) (2, -3) (6, -3) (-3, -3) (7, -3) 5 5 3 (2, -6)

Graph the following Ellipse (-1, 12) Center: (-1, 5) 7 a = 7 in y direction (-1, 5) (-5, 5) (3, 5) 4 4 b = 4 in x direction 7 (-1, -2)

Graph the following Ellipse a = 7 b = 4 a2 = b2 + c2 72 = 42 + c2 49 = 16 + c2 (-1, 12) 33 = c2 foci 7 5.7 = c (-1, 5) (-5, 5) (3, 5) 4 4 7 (-1, -2)

Graph the following Ellipse Center: (-1, -3) a = 3 in x direction b = 2 in y direction (-1, -1) 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5)

Graph the following Ellipse a = 3 b = 2 a2 = b2 + c2 32 = 22 + c2 9 = 4 + c2 foci 5 = c2 (-1, -1) 2.2 = c 2 (-1, -3) (-4, -3) (2, -3) 3 3 2 (-1, -5)

Graph the following Ellipse Center: (-2, 5) a = 9 in y direction b = 6 in x direction (-2, 14) 9 (-2, 5) (-8, 5) (4, 5) 6 6 9 (-2, -4)

Graph the following Ellipse a = 9 b = 6 a2 = b2 + c2 92 = 62 + c2 81 = 36 + c2 (-2, 14) 45 = c2 foci 9 6.7 = c (-2, 5) (-8, 5) (4, 5) 6 6 9 (-2, -4)

Graph the following Ellipse Center: (4, 3) a = 5 in x direction b = 2 in y direction (4, 5) 2 (4, 3) (-1, 3) (9, 3) 5 5 2 (4, 1)

Graph the following Ellipse a = 5 b = 2 a2 = b2 + c2 52 = 22 + c2 25 = 4 + c2 foci 21 = c2 (4, 5) 4.6 = c 2 (4, 3) (-1, 3) (9, 3) 5 5 2 (4, 1)