Chapter 8: Problem 1
Find \(d y / d x\). $$ x=t^{2}, y=5-4 t $$
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In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{6}{2+\cos \theta}\)
In Exercises 57 and \(58,\) let \(r_{0}\) represent the distance from the focus to the nearest vertex, and let \(r_{1}\) represent the distance from the focus to the farthest vertex. Show that the eccentricity of an ellipse can be written as \(e=\frac{r_{1}-r_{0}}{r_{1}+r_{0}} .\) Then show that \(\frac{r_{1}}{r_{0}}=\frac{1+e}{1-e}\).
Describe what happens to the distance between the directrix and the center of an ellipse if the foci remain fixed and \(e\) approaches 0 .
Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=e^{2 t}, \quad y=e^{t} $$
Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=t-1, \quad y=\frac{t}{t-1} $$
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