Chapter 10: Problem 6
What is the polar equation of the horizontal line \(y=5 ?\)
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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. The length of the latus rectum of the parabola \(y^{2}=4 p x\) or \(x^{2}=4 p y\) is \(4|p|\)
A plane traveling horizontally at \(80 \mathrm{m} / \mathrm{s}\) over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by $$x=80 t, \quad y=-4.9 t^{2}+3000, \quad \text { for } t \geq 0$$ where the origin is the point on the ground directly beneath the plane at the moment of the release. Graph the trajectory of the packet and find the coordinates of the point where the packet lands.
Use a graphing utility to graph the hyperbolas \(r=\frac{e}{1+e \cos \theta},\) for \(e=1.1,1.3,1.5,1.7\) and 2 on the same set of axes. Explain how the shapes of the curves vary as \(e\) changes.
Explain and carry out a method for graphing the curve \(x=1+\cos ^{2} y-\sin ^{2} y\) using parametric equations and a graphing utility.
Show that the set of points equidistant from a circle and a line not passing through the circle is a parabola. Assume the circle, line, and parabola lie in the same plane.
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