Exam 10: Parametric Equations and Polar Coordinates

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. 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.

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 $\left( 1 , \frac { \pi } { 2 } \right)$ .

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

Sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $x = \cos \theta , y = 5 \sec \theta , 0 \leq \theta < \frac { \pi } { 2 }$

Describe the motion of a particle with position $( x , y )$ as $t$ varies in the given interval $0 \leq t \leq 2 \pi$ $x = 8 \sin t , y = 5 \cos t$

Find an equation of the ellipse that satisfies the given conditions. Foci: $( 0 , \pm 8 )$ , vertices $( 0 , \pm 9 )$

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

The graph of the following curve is given. Find the area that it encloses. $r = 3 + 15 \sin 6 \theta$

Find the area of the region that lies inside both curves. $r = 8 + 2 \sin \theta , r = 7$

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

$\text { Find parametric equations to represent the line segment from } ( - 3,4 ) \text { to } ( 12 , - 8 ) \text {. }$

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

Find the vertices, foci and asymptotes of the hyperbola. $5 x ^ { 2 } - 5 y ^ { 2 } + 40 x - 50 y = 70$

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

Eliminate the parameter to find a Cartesian equation of the curve. $x ( t ) = 2 \cos ^ { 2 } t , \quad y ( t ) = 7 \sin ^ { 2 } t$

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 $x = - 13$ .

Find the area that the curve encloses. $r = 13 \sin \theta$

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.

The orbit of Hale-Bopp comet, discovered in 1995 , is an ellipse with eccentricity $0.995$ and one focus at the Sun. The length of its major axis is $366.5 \mathrm { AU }$ . [An astronomical unit (AU) is the mean distance between Earth and the Sun, about 93 million miles.] Find the maximum distance from the comet to the Sun. (The perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ .) Find the answer in $\mathrm { AU }$ and round to the nearest hundredth.