Convert the polar equation to rectangular form.

$$r=8\sin\theta$$

Graphical Reasoning Consider the parabola $x^2=4 p y$

For each parabola in part (a), find the length of the latus recturn (see figure). How can the length of the latus rectum be determined directly from the standard form of the equation of the parabola?

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Use a graphing utility to graph the equation of the orbit.

Graphical Reasoning Consider the parabola $x^2=4 p y$

Locate the focus for each parabola in part (a).

Graphical Reasoning Consider the parabola $x^2=4py$.

Use a graphing utility to graph the parabola for $p=1,p=2,p=3,\ \text{and}, p=4$ and Describe the effect on the graph when increases.

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the $x$-axis.

Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=2t,\ y=3t-1$$

$$\underline{Point}$$

$$t=3$$

Convert the rectangular equation to polar form. Assume $a>0$.

$$y^3=x^2$$

Convert the polar equation to rectangular form.

$$r=3\cos\theta$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

$$4x^2-6x-2y+7=0$$

Find $dy/dx$.

$$x=2e^\theta,y=e^{-\theta/2}$$

Convert the rectangular equation to polar form. Assume $a>0$.

$$x^2+y^2-2ay=0$$

Satellite Orbit The first artificial satellite to orbit Earth was Sputnik I (launched by the former Soviet Union in $195$7). Its highest point above Earth’s surface was $947$ kilometers, and its lowest point was $228$ kilometers. The center of Earth was at one focus of the elliptical orbit, and the radius of Earth is $6378$ kilometers. Find the eccentricity of the orbit..

Describe in words how a plane could intersect with the double-napped cone shown to form the conic section.

Hyperbola