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Stitz-Zeager_College_Algebra_e-book

Find the distance in au between the sun and aphelion

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Unformatted text preview: le. 398 Hooked on Conics Tilting the plane ever so slightly produces an ellipse. If the plane cuts parallel to the cone, we get a parabola. If we slice the cone with a vertical plane, we get a hyperbola. For a wonderful animation describing the conics as intersections of planes and cones, see Dr. Louis Talman’s Mathematics Animated Website. 7.1 Introduction to Conics 399 If the slicing plane contains the vertex of the cone, we get the so-called ‘degenerate’ conics: a point, a line, or two intersecting lines. We will focus the discussion on the non-degenerate cases: circles, parabolas, ellipses, and hyperbolas, in that order. To determine equations which describe these curves, we will make use of their definitions in terms of distances. 400 7.2 Hooked on Conics Circles Recall from geometry that a circle can be determined by fixing a point (called the center) and a positive number (called the radius) as follows. Definition 7.1. A circle with center (h, k ) and radius r > 0 is the set of all points (x, y )...
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