Exam 10: Parametric Equations and Polar Coordinates

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $x = t - 2 \sin t , \quad y = 1 - 2 \cos t , \quad 0 \leq t \leq 5 \pi$

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

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

Match the equation with the correct graph. $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$

Write a polar equation of the conic that has a focus at the origin, eccentricity $\frac { 3 } { 2 }$ , and directrix $x = - 6$ . Identify the conic.

Find an equation of the parabola with focus $\left( \frac { 15 } { 2 } , 0 \right)$ and directrix $x = - \frac { 13 } { 2 }$ .

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

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 $106$ km and apolune altitude $318$ km (above the moon). Find an equation of this ellipse if the radius of the moon is $1728$ km and the center of the moon is at one focus.

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

Consider the polar equation $r = \frac { 5 } { 3 - 3 \cos \theta }$ . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Find an equation of the tangent to the curve at the point by first eliminating the parameter. $x = e ^ { t }$ , $y = ( t - 7 ) ^ { 2 }$ ; $( 1,64 )$

Find an equation of the conic satisfying the given conditions. Hyperbola, foci (5, 6) and (5, -4), asymptotes x = 2y + $3$ and x = - 2y + $7$

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 L = $440$ mi due east of station A on a coastline. A ship received the signal from B $1280$ microseconds (µs) before it received the signal from A. Assuming that radio signals travel at a speed of $980$ ft /µs and if the ship is due north of B, how far off the coastline is the ship? Round your answer to the nearest mile.

Find an equation of the hyperbola with vertices $( 0 , \pm 6 )$ and asymptotes $y = \pm \frac { x } { 3 }$ .

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

The planet Mercury travels in an elliptical orbit with eccentricity $0.403$ . Its minimum distance from the Sun is $6.9 \times 10 ^ { 7 }$ km. If the perihelion distance from a planet to the Sun is $a ( 1 - e )$ and the aphelion distance is $a ( 1 + e )$ , find the maximum distance (in km) from Mercury to the Sun.

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

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

Find the point(s) of intersection of the curves $r = 2$ and $r = 4 \cos \theta$ .

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