Exam 10: Parametric Equations and Polar Coordinates

Find the eccentricity of the conic. Select the correct answer. $r = \frac { 5 } { 8 - 5 \sin \theta }$

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $\frac { 6 } { 7 }$ and directrix $x = - 13$ .

Write a polar equation in $\mathrm { r }$ and $\theta$ of a hyperbola with the focus at the origin, with the eccentricity 7 and directrix $r = - 12 \csc \theta$ .

Find the polar equation for the curve represented by the given Cartesian equation. $x + y = 2$

Write a polar equation in $r$ and $\theta$ of an ellipse with the focus at the origin, with the eccentricity $0.8$ and vertex at $\left( 1 , \frac { \pi } { 2 } \right)$ .

Sketch the polar curve with the given equation. $r = \sin 2 \theta , \quad - \pi \leq x \leq \pi$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ . $x = 4 ( t + \sin t ) , y = 4 ( t - \cos t )$

Suppose a planet is discovered that revolves around its sun in an elliptical orbit with the sun at one focus. Its perihelion distance (minimum distance from the planet to the sun) is approximately $2.3 \times 10 ^ { 7 } \mathrm {~km}$ , and its aphelion distance (maximum distance from the planet to the sun) is approximately $2.7 \times 10 ^ { 7 } \mathrm {~km}$ . Approximate the eccentricity of the planet's orbit. Round to three decimal places.

Find the length of the polar curve. $r = 3 \cos \theta , 0 \leq \theta \leq \frac { 3 \pi } { 4 }$

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude $106 \mathrm {~km}$ and apolune altitude $318 \mathrm {~km}$ (above the moon). Find an equation of this ellipse if the radius of the moon is $1728 \mathrm {~km}$ and the center of the moon is at one focus.

Find the exact area of the surface obtained by rotating the given curve about the $x$ -axis. $x = 2 \cos ^ { 3 } \theta , \quad y = 2 \sin ^ { 3 } \theta , \quad 0 \leq \theta \leq \pi / 2$

Find a polar equation for the curve represented by the given Cartesian equation. Select the correct answer. $x ^ { 2 } = 3 y$

Find an equation for the conic that satisfies the given conditions. hyperbola, foci $( 0 , \pm 6 )$ , vertices $( 0 , \pm 3 )$

Graph of the following curve is given. Find its length. Select the correct answer. $r = 6 \cos ^ { 2 } \left( \frac { \theta } { 2 } \right)$

Find an equation of the ellipse that satisfies the given conditions. Foci: $( 0 , \pm 1 )$ , vertices $( 0 , \pm 6 )$

Find an equation of the conic satisfying the given conditions. Hyperbola, foci $( 5,6 )$ and $( 5 , - 2 )$ , asymptotes $x = 2 y + 1$ and $x = - 2 y + 9$

In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at $A$ and $B$ transmit simultaneous signals to a ship or an aircraft located at $P$ . The onboard computer converts the time difference in receiving these signals into a distance difference $| A | - | B |$ , and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure). Suppose that station $B$ is located $\mathrm { L } = 480 \mathrm { mi }$ due east of station $A$ on a coastline. $\mathrm { A }$ ship received the signal from $B 1280$ microseconds $( \mu s )$ before it received the signal from $A$ . Assuming that radio signals travel at a speed of $1000 \mathrm { ft } / \mu \mathrm { s }$ and if the ship is due north of $B$ , how far off the coastline is the ship? Round your answer to the nearest mile. Select the correct answer.