Placing station c yields the hyperbola 36 2464 at 150

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Unformatted text preview: n pictures we have, Hooked on Conics Minor Axis 420 Major Axis V1 F1 F2 C V2 An ellipse with center C ; foci F1 , F2 ; and vertices V1 , V2 Note that the major axis is the longer of the two axes through the center, and likewise, the minor axis is the shorter of the two. In order to derive the standard equation of an ellipse, we assume that the ellipse has its center at (0, 0), its major axis along the x-axis, and has foci (c, 0) and (−c, 0) and vertices (−a, 0) and (a, 0). We will label the y -intercepts of the ellipse as (0, b) and (0, −b) (We assume a, b, and c are all positive numbers.) Schematically, y (0, b) (x, y ) x (−a, 0) (−c, 0) (c, 0) (a, 0) (0, −b) Note that since (a, 0) is on the ellipse, it must satisfy the conditions of Definition 7.4. That is, the distance from (−c, 0) to (a, 0) plus the distance from (c, 0) to (a, 0) must equal the fixed distance d. Since all of these points lie on the x-axis, we get distance from (−c, 0) to (a, 0) + distance from (c, 0) to (a, 0) = d (a + c) + (a − c) = d 2a = d 7.4 Ellipses 421 In other words, the fixed distance d mentioned in the definition of the ellipse is none other than the length of the major axis. We now use...
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