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Engineering Calculus Notes 149

Engineering Calculus Notes 149 - 2.2 Parametrized Curves...

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2.2. PARAMETRIZED CURVES 137 B C F A E G D Figure 2.13: Trisecting an Angle 8. Show that the sum of the distances from a point on an ellipse to its two foci equals the major axis. (You may assume the equation is in standard form.) This is sometimes called the Gardener’s characterization of an ellipse: explain how one can construct an ellipse using a piece of string. 9. Show that the (absolute value of the) difference between the distances from a point on a hyperbola to its two foci equals the transverse axis. (You may assume the equation is in standard form.) 10. Show that the locus of the equation xy = 1 is a hyperbola. ( Hint: consider a different coordinate system, using the diagonal and anti-diagonal as axes.)
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Unformatted text preview: 2.2 Parametrized Curves Parametrized Curves in the Plane There are two distinct ways of specifying a curve in the plane. In classical geometric studies, a curve is given in a static way, either as the intersection of the plane with another surface (like the conical surface in Apollonius) or by a geometric condition (like ²xing the distance from a point or the focus-directrix property in Euclid and Pappus). This approach reached its modern version in the seventeenth century with Descartes’ and Fermat’s formulation of a curve as the locus of an equation in the coordinates of a...
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