91Ó°ÊÓ

If the graph of the equation is an ellipse, find the coordinates of the endpoints of the minor axis. If the graph of the equation is a hyperbola, find the equations of the asymptotes. If the graph of the equation is a parabola, find the coordinates of the vertex. Express answers relative to an \(x^{\prime} y^{\prime}\) -system in which the given equation has no \(x^{\prime} y^{\prime}\) -term. Assume that the \(x^{\prime} y^{\prime}\) -system has the same origin as the \(x y\) -system. $$2 x^{2}-4 x y+5 y^{2}-36=0$$

In Exercises \(41-43\), eliminate the parameter. Write the resulting equation in standard form. An ellipse: \(x=h+a \cos t, y=k+b \sin t\)

Use the polar equation for planetary orbits, $$r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta}$$ to find the polar equation of the orbit for Mercury and Earth. Mercury: \(e=0.2056\) and \(a=36.0 \times 10^{6}\) miles Earth: \(\quad e=0.0167\) and \(a=92.96 \times 10^{6}\) miles Use a graphing utility to graph both orbits in the same viewing rectangle. What do you see about the orbits from their graphs that is not obvious from their equations?

Graph each ellipse and give the location of its foci. $$\frac{(x+1)^{2}}{2}+\frac{(y-3)^{2}}{5}=1$$

Write a polar equation of the conic that is named and described. Ellipse: a focus at the pole; vertex: \((4,0) ; e=\frac{1}{2}\)

Identify the conic and write its equation in rectangular coordinates: \(r=\frac{1}{2-2 \cos \theta}\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. In order to graph an ellipse whose equation contained an \(x y\) -term, I used a rotated coordinate system that placed the ellipse's center at the origin.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Although the algebra of rotations can get ugly, the main idea is that rotation through an appropriate angle will transform a general second-degree equation into an equation in \(x^{\prime}\) and \(y^{\prime}\) without an \(x^{\prime} y^{\prime}\) -term.

An explosion is recorded by two microphones that are 1 mile apart. Microphone \(M_{1}\) received the sound 2 seconds before microphone \(M_{2} .\) Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones.

In Exercises \(63-68\), sketch the function represented by the given parametric equations. Then use the graph to determine each of the following: a. intervals, if any, on which the function is increasing and intervals, if any, on which the function is decreasing. b. the number, if any, at which the function has a maximum and this maximum value, or the number, if any, at which the function has a minimum and this minimum value. $$x=\frac{t}{2}, y=2 t^{2}-8 t+3$$