Question 1

The planet Mercury travels in an elliptical orbit with eccentricity  . Its minimum distance from the Sun is  km. If the perihelion distance from a planet to the Sun is  and the aphelion distance is  , find the maximum distance (in km) from Mercury to the Sun.
A)  km
B)  km
C)  km
D)  km
E)  km

Accepted Answer

E

Question 2

Find an equation of the hyperbola with vertices  and asymptotes  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

A

Question 3

Find an equation for the conic that satisfies the given conditions. hyperbola, foci (0, ±  ) , vertices (0, ±  )
A) 
B) 
C) 
D) 
E)

Accepted Answer

D

Question 4

Find the eccentricity of the conic. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the eccentricity of the conic. 
A)...

Question 5

Match the equation with the correct graph. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Match the equation with the correct graph....

Question 6

Write a polar equation in r and  of a hyperbola with the focus at the origin, with the eccentricity  and directrix  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write a polar equation in r and...

Question 7

Consider the polar equation  .
(a) Find the eccentricity and an equation of the directrix of the conic.
(b) Identify the conic.
(c) Sketch the curve.

Accepted Answer

The answer of Consider the polar equation  .
(a) Find...

Question 8

Write a polar equation of the conic that has a focus at the origin, eccentricity  , and directrix  . Identify the conic.
A)  , ellipse
B)  , ellipse
C)  , hyperbola
D)  , hyperbola

Accepted Answer

A polar equation for a hyperbola with focus at the origin and directrix in polar form is given by $r=\frac{ed}{1+e\cos{	heta}}$, where $e$ is the eccentricity, $d$ is the distance from the origin to the directrix, and $	heta$ is the polar angle. Plugging in the given values, we get $r=\frac{11}{1+11\cos{	heta}}$. This is the equation of a hyperbola. Therefore, the answer is C.

Question 9

Find the equation of the directrix of the conic. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The directrix of a conic with eccentricity $e$ and focus $(h,k)$ is the line perpendicular to the major axis and a distance of $\frac{1}{e}$ units away from the focus. In this case, we have $e=\frac{1}{6}$ and the focus is at $(0,0)$ (since the conic has equation $6x^2-6y^2=1$). Therefore, the directrix is a vertical line $x=\frac{1}{e}=6$. The equation of this line is $x-6=0$, which can be written as $11eaa3e5\_5519\_be05\_9f8f\_8fca8e7ccde7\_TB5971\_11$.

Question 10

Write a polar equation in r and $\theta$of an ellipse with the focus at the origin, with the eccentricity  and vertex at  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write a polar equation in r and...

Question 11

Find an equation for the conic that satisfies the given conditions. ellipse, foci  , length of major axis 8
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find an equation for the conic that...

Question 12

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  , 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 =  mi due east of station A on a coastline. A ship received the signal from B  microseconds (µs) before it received the signal from A. Assuming that radio signals travel at a speed of  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. 
A)  miles
B)  miles
C)  miles
D)  miles
E)  miles

Accepted Answer

The answer of In the LORAN (LOng RAnge Navigation) radio...

Question 13

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0), asymptotes: y = ±  x
A) 
B) 
C) 
D)

Accepted Answer

The answer of Find an equation of the hyperbola centered...

Question 14

Find an equation for the conic that satisfies the given conditions. parabola, vertex (0, 0), focus (0, -  )
A) 
B) 
C) 
D) 
E)

Accepted Answer

Since the vertex is at (0,0), we know that the axis of symmetry is the y-axis. This means that the parabola is of the form $x^2=4py$ for some value of $p$ (the distance from the vertex to the focus). We are given that the focus is at $(0,-11eaa3e5_551a_cf93_9f8f_ab3f2305aa20_TB5971_11)$, which means that $p=11eaa3e5_551a_cf93_9f8f_ab3f2305aa20_TB5971_11$. Substituting this value for $p$ into the equation of the parabola, we get $x^2=44eaa3e5_551a_cf93_9f8f_ab3f2305aa20_TB5971_11y$. Simplifying the coefficient gives the answer of choice B.

Question 15

The orbit of Hale-Bopp comet, discovered in 1995, is an ellipse with eccentricity  and one focus at the Sun. The length of its major axis is  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  and the aphelion distance is  .) Find the answer in AU and round to the nearest hundredth.
A)  AU
B)  AU
C)  AU
D)  AU
E)  AU

Accepted Answer

The answer of The orbit of Hale-Bopp comet, discovered in...

Question 16

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 1.3  km, and its aphelion distance (maximum distance from the planet to the sun) is approximately 6.9  km. Approximate the eccentricity of the planet's orbit. Round to three decimal places.
A)0.683
B)1.464
C)5.308
D)0.188

Accepted Answer

The answer of Suppose a planet is discovered that revolves...

Question 17

Write a polar equation in r and $\theta$ of an ellipse with the focus at the origin, with the eccentricity  and directrix  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write a polar equation in r and...

Question 18

Find an equation of the parabola with focus  and directrix  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The vertex of the parabola is the midpoint between the focus and the directrix. Therefore, the vertex has coordinates $(0, \frac{1}{2}(0+4))=(0,2)$. Since the directrix is a horizontal line, the parabola opens upwards. Let $p$ be the distance between the vertex and the focus. This is also the distance between the vertex and the directrix. Thus, $p=2$. Using the definition of a parabola, we have $(x-0)^2+(y-2)^2=(2)^2+((y-4)^2$. Simplifying gives $x^2+(y-2)^2=4+(y-4)^2$, which expands to $x^2+y^2-4y+0x-8=0$. Completing the square gives $(x-0)^2+(y-2)^2=16$, which is the standard form of a parabola with vertex $(0,2)$ and $p=2$. Thus, the equation of the parabola is $x^2+(y-2)^2=16-y^2+4y$. Distributing the negative $y^2$ and combining like terms gives $x^2+2y^2-4y-16=0$. The only choice that matches this answer is $\boxed{	extbf{(B)}}$

Question 19

Consider the polar equation  .
(a) Find the eccentricity and an equation of the directrix of the conic.
(b) Identify the conic.
(c) Sketch the curve.

Accepted Answer

The answer of Consider the polar equation  .
(a) Find...

Question 20

Consider the polar equation  .
(a) Find the eccentricity and an equation of the directrix of the conic.
(b) Identify the conic.
(c) Sketch the curve.

Accepted Answer

The answer of Consider the polar equation  .
(a) Find...

Quiz 10: Parametric Equations and Polar Coordinates

The planet Mercury travels in an elliptical orbit with eccentricity . Its minimum distance from the Sun is km. If the perihelion distance from a planet to the Sun is and the aphelion distance is , find the maximum distance (in km) from Mercury to the Sun.

Find an equation of the hyperbola with vertices and asymptotes .

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

Find the eccentricity of the conic.

Match the equation with the correct graph.

Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and directrix .

Consider the polar equation . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Write a polar equation of the conic that has a focus at the origin, eccentricity , and directrix . Identify the conic.

Find the equation of the directrix of the conic.

Find an equation for the conic that satisfies the given conditions. ellipse, foci , length of major axis 8

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0) , asymptotes: y = ± x

Find an equation for the conic that satisfies the given conditions. parabola, vertex (0, 0) , focus (0, - )

Find an equation of the parabola with focus and directrix .

Consider the polar equation . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Consider the polar equation . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Quiz 10: Parametric Equations and Polar Coordinates

The planet Mercury travels in an elliptical orbit with eccentricity . Its minimum distance from the Sun is km. If the perihelion distance from a planet to the Sun is and the aphelion distance is , find the maximum distance (in km) from Mercury to the Sun.

Find an equation of the hyperbola with vertices and asymptotes .

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

Find the eccentricity of the conic.

Match the equation with the correct graph.

Write a polar equation in r and of a hyperbola with the focus at the origin, with the eccentricity and directrix .

Consider the polar equation . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Write a polar equation of the conic that has a focus at the origin, eccentricity , and directrix . Identify the conic.

Find the equation of the directrix of the conic.

Write a polar equation in r and θ\thetaθ of an ellipse with the focus at the origin, with the eccentricity and vertex at .

Find an equation for the conic that satisfies the given conditions. ellipse, foci , length of major axis 8

Find an equation of the hyperbola centered at the origin that satisfies the given condition. Vertices: (± 4, 0) , asymptotes: y = ± x

Find an equation for the conic that satisfies the given conditions. parabola, vertex (0, 0) , focus (0, - )

Write a polar equation in r and θ\thetaθ of an ellipse with the focus at the origin, with the eccentricity and directrix .

Find an equation of the parabola with focus and directrix .

Consider the polar equation . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.

Consider the polar equation . (a) Find the eccentricity and an equation of the directrix of the conic. (b) Identify the conic. (c) Sketch the curve.