91Ó°ÊÓ

Finding the Standard Equation of a a Hyperbola In Exercises \(41-48,\) find the standard form of the equation of the hyperbola with the given characteristics. \(\begin{aligned} \text { Vertices: }(0,2), &(6,2) \\ \text { Asymptotes: } y &=\frac{2}{3} x \\ y &=4-\frac{2}{3} x \end{aligned}\)

Arc Length In Exercises 49-54, find the arc length of the curve on the given interval. $$\begin{array}{ll}{\text { Parametric Equations }} & {\text { Interval }} \\\ {x=6 t^{2}, \quad y=2 t^{3}} & {1 \leq t \leq 4}\end{array}$$

Area Sketch the strophoid \(r=\sec \theta-2 \cos \theta, \quad-\frac{\pi}{2}<\theta<\frac{\pi}{2}\) Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

Finding the Arc Length of a Polar Curve In Exercises \(53-58\) , find the length of the curve over the given interval. \(r=8, \quad\left[0, \frac{\pi}{6}\right]\)

Planetary Motion The planets travel in elliptical orbits with the sun as a focus, as shown in the figure. (a) Show that the polar equation of the orbit is given by \(r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}\) where \(e\) is the eccentricity. (b) Show that the minimum distance (perihelion) from the sun to the planet is \(r=a(1-e)\) and the maximum distance (aphelion) is \(r=a(1+e)\) .

Folium of Descartes Consider the parametric equations $$x=\frac{4 t}{1+t^{3}} \quad \text { and } \quad y=\frac{4 t^{2}}{1+t^{3}}$$ $$\begin{array}{l}{\text { (a) Use a graphing utility to graph the curve represented by }} \\ {\text { the parametric equations. }} \\ {\text { (b) Use a graphing utility to find the points of horizontal }} \\ {\text { tangency to the curve. }}\end{array}$$ $$\begin{array}{l}{\text { (c) Use the integration capabilities of a graphing utility to }} \\ {\text { approximate the arc length of the closed loop. (Hint: Use }} \\ {\text { symmetry and integrate over the interval } 0 \leq t \leq 1 .}\end{array}$$

Proof (a) Prove that if any two tangent lines to a parabola intersect at right angles, then their point of intersection must lie on the directrix. (b) Demonstrate the result of part (a) by showing that the tangent lines to the parabola \(x^{2}-4 x-4 y+8=0\) at the points \((-2,5)\) and \(\left(3, \frac{5}{4}\right)\) intersect at right angles and that their point of intersection lies on the directrix.

Comet Hale-Bopp The comet Hale-Bopp has an elliptical orbit with the sun at one focus and has an eccentricity of \(e \approx 0.995 .\) The length of the major axis of the orbit is approximately 500 astronomical units. (a) Find the length of its minor axis. (b) Find a polar equation for the orbit. (c) Find the perihelion and aphelion distances.

Satellite Orbit The apogee (the point in orbit farthest from Earth) and the perigee (the point in orbit closest to Earth) of an elliptical orbit of an Earth satellite are given by \(A\) and \(P\) respectively. Show that the eccentricity of the orbit is \(e=\frac{A-P}{A+P}\)

Explorer 55 On November \(20,1975,\) the United States launched the research satellite Explorer \(55 .\) Its low and high points above the surface of Earth were 96 miles and 1865 miles. Find the eccentricity of its elliptical orbit. (Use 4000 miles as the radius of Earth.)