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Chapter 6: Problem 40
Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: \((0, \pm 3) ;\) asymptotes: \(y=\pm 3 x\)
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(a) Show that the distance between the points \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) is \(\sqrt{r_{1}^{2}+r_{2}^{2}-2 r_{1} r_{2} \cos \left(\theta_{1}-\theta_{2}\right)}\) (b) Simplify the Distance Formula for \(\theta_{1}=\theta_{2} .\) Is the simplification what you expected? Explain. (c) Simplify the Distance Formula for \(\theta_{1}-\theta_{2}=90^{\circ}\) Is the simplification what you expected? Explain.
In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{1}{1-\cos \theta}$$
Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form.
Verifying a Polar Equation Show that the polar equation of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad\) is \(\quad r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta}\)
A projectile is launched at a height of \(h\) feet above the ground at an angle of \(\theta\) with the horizontal. The initial velocity is \(v_{0}\) feet per second, and the path of the projectile is modeled by the parametric equations $$x=\left(v_{0} \cos \theta\right) t$$ and $$y=h+\left(v_{0} \sin \theta\right) t-16 t^{2}.$$ Use a graphing utility to graph the paths of a projectile launched from ground level at each value of \(\boldsymbol{\theta}\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and the range of the projectile. (a) \(\theta=60^{\circ}, \quad v_{0}=88\) feet per second (b) \(\theta=60^{\circ}, \quad v_{0}=132\) feet per second (c) \(\theta=45^{\circ}, \quad v_{0}=88\) feet per second (d) \(\theta=45^{\circ}, \quad v_{0}=132\) feet per second
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