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ss_9_3

# ss_9_3 - Section 9.3 The Ellipse Recall that an ellipse can...

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Section 9.3 The Ellipse Recall that an ellipse can be obtained as the graph of the quadratic equation Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0 with discriminant B 2 - 4 AC < 0. An alternate method for obtaining an ellipse is given by the following deﬁnition. Let F 1 and F 2 be any two distinct points in the plane. An ellipse is the set of all points P in the plane such that d ( P, F 1 )+ d ( P, F 2 ) = constant, where d ( A, B ) denotes the distance from A to B. The points F 1 and F 2 are called the foci of the ellipse. The graph of an ellipse is shown below. In the above ﬁgure, the points V 1 and V 2 are called vertices of the ellipse. The line segment between V 1 and V 2 is called the major axis . The line segment through the center of the ellipse and perpendicular to the major axis is called the minor axis . The equation of an ellipse is much simpler when its major axis is either horizontal or vertical. Ellipses centered at the origin and having horizontal and vertical axes are shown below. Note in these diagrams the relationship between the distances a , b ,and c . The number a is half the length of the major axis, the number b is half the length of the minor axis, and the number c is half the distance between the foci. Hence, a>b a>c .Ineachcase a 2 = b 2 + c 2 . x 2 a 2 + y 2 b 2 =1 y 2 a 2 + x 2 b 2 Remark. In the equation x 2 r 2 + y 2 s 2 = 1, the larger of r, s is the number a and the smaller of r, s is the number b . The larger of the two numbers r , s determines whether the major axis is either horizontal or vertical.

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ss_9_3 - Section 9.3 The Ellipse Recall that an ellipse can...

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