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Chapter 10: Problem 11
What are the equations of the asymptotes of a standard hyperbola with vertices on the \(x\) -axis?
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Find the equation in Cartesian coordinates of the lemniscate \(r^{2}=a^{2} \cos 2 \theta,\) where \(a\) is a real number.
Slopes of tangent lines Find all the points at which the following curves have the given slope. $$x=2 \cos t, y=8 \sin t ; \text { slope }=-1$$
A focal chord of a conic section is a line through a focus joining two points of the curve. The latus rectum is the focal chord perpendicular to the major axis of the conic. Prove the following properties. Let \(L\) be the latus rectum of the parabola \(y^{2}=4 p x,\) for \(p>0\) Let \(F\) be the focus of the parabola, \(P\) be any point on the parabola to the left of \(L,\) and \(D\) be the (shortest) distance between \(P\) and \(L\) Show that for all \(P, D+|F P|\) is a constant. Find the constant.
Consider an ellipse to be the set of points in a plane whose distances from two fixed points have a constant sum 2 \(a .\) Derive the equation of an ellipse. Assume the two fixed points are on the \(x\) -axis equidistant from the origin.
Find parametric equations (not unique) of the following ellipses (see Exercises \(75-76\) ). Graph the ellipse and find a description in terms of \(x\) and \(y.\) An ellipse centered at (-2,-3) with major and minor axes of lengths 30 and \(20,\) parallel to the \(x\) - and \(y\) -axes, respectively, generated counterclockwise (Hint: Shift the parametric equations.)
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