Question 1

Write the equation of the ellipse with foci at (4, 2), (6, 2), and b = 2.
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Write the equation of the ellipse with...

Question 2

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 4).
A) 
B) 
C) 
D) 
E)none of these

Accepted Answer

The answer of Find the equation of the parabola with...

Question 3

Find the equation of the parabola with vertex at (3, 4) and focus at (-4, 4).
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the equation of the parabola with...

Question 4

Find the graph of the following ellipse. 
A) 
B) 
C) 
D)

Accepted Answer

The graph of an ellipse is determined by its center, semi-major and semi-minor axes, and orientation. The center of this ellipse is at the origin, which can be seen by noting that the equation is in the form (x^2)/(a^2) + (y^2)/(b^2) = 1. The values of a and b are 2 and 1, respectively. Finally, the ellipse is oriented along the x-axis because a > b. These properties match those of the graph in option B.

Question 5

Find the vertex and directrix of the parabola. 
A)vertex:  directrix: 
B)vertex:  directrix: 
C)vertex:  directrix: 
D)vertex:  directrix: 
E)vertex:  directrix:

Accepted Answer

The answer of Find the vertex and directrix of the...

Question 6

Find the standard form of the equation of the hyperbola with the given characteristics. vertices:  foci: 
A) 
B) 
C) 
D) 
E)

Accepted Answer

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Question 7

Graph the hyperbola. 
A) 
B) 
C) 
D)

Accepted Answer

The answer of Graph the hyperbola. 
A) 
B) 
C) 
D)...

Question 8

Give the coordinates of the circle's center and its radius. 
A)(0, 0); r = 144
B)(0, 0); r = 12
C)(1, 1); r = 144
D)(1, 1); r = 12
E)none of these

Accepted Answer

The answer of Give the coordinates of the circle's center...

Question 9

In a suspension bridge, the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 400 m apart, and the lowest point of the suspension cables is 100 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the lowest point of the cable. NOTE: This equation is used to find the length of the cable needed in the construction of the bridge. A)x² = 800y B)x² = 400y C)x² = -400y D)x² = 1,600y E)y² = 1,600x

Accepted Answer

Since the towers are 400m apart, the distance from the origin to each tower is 200m. Also, since the lowest point of the cable is 100m below the top of the towers, the vertex of the parabola is at (0,-100). Therefore, the equation of the parabolic part of the cable can be written in the form:

y = a(x-0)^2 - 100

We know that the parabola passes through the two towers, which are located at (-200,0) and (200,0). Plugging these points into the equation gives us:

0 = a(-200-0)^2 - 100 
0 = a(200-0)^2 - 100

Simplifying these equations gives us:

a = 1/800

Substituting this value of a into the equation y = a(x-0)^2 - 100 and simplifying gives us:

y = (1/800)x^2 - 100

Multiplying both sides by 800 gives the final equation:

x^2 = 400y, which can be written as 2S1S1P0 = 400y, where S1 is the midpoint of the line segment connecting the two towers. Therefore, the correct choice is B.

Question 10

Identify the graph of the following ellipse. 
A) 
B) 
C)

Accepted Answer

The graph in choice C has a vertical major axis and a center at (4, -1), which matches the given equation.

Question 11

Find the vertices and asymptotes of the hyperbola. 
A)vertices:  asymptote: 
B)vertices:  asymptote: 
C)vertices:  asymptote: 
D)vertices:  asymptote: 
E)vertices:  asymptote:

Accepted Answer

The answer of Find the vertices and asymptotes of the...

Question 12

Find the standard form of the equation of the ellipse with vertices  and eccentricity  .
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the standard form of the equation...

Question 13

Find the equation of the circle graphed below. 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The answer of Find the equation of the circle graphed...

Question 14

Find the center and vertices of the ellipse. 
A)center:  vertices: 
B)center:  vertices: 
C)center:  vertices: 
D)center:  vertices: 
E)center:  vertices:

Accepted Answer

The answer of Find the center and vertices of the...

Question 15

Write the equation of the ellipse that has its center at the origin with focus at (0, 5) and vertex at (0, 10).
A) 
B) 
C) 
D) 
E)none of these

Accepted Answer

Since the center is at the origin, we know that the equation of the ellipse is of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. We also know that the distance from the center to the focus is $c=5$, and the distance from the center to the vertex is $a=10$. We can use the formula $c^2=a^2-b^2$ to solve for $b$:
$$5^2=10^2-b^2 \Rightarrow b^2=75$$
Plugging in $a=10$ and $b=\sqrt{75}$, we get the equation:
$$\frac{x^2}{10^2}+\frac{y^2}{\sqrt{75}^2}=1$$
Simplifying and rationalizing the denominator, we get the equation:
$$\frac{x^2}{100}+\frac{y^2}{25}=1$$
So the answer is B.

Question 16

Any sound from one focus of an ellipse reflects off the ellipse directly back to the other focus. This property explains whispering galleries such as Statuary Hall in Washington, D.C. The whispering gallery shown in illustration has the shape of a semi-ellipse. Find the distance sound travels as it leaves focus F and returns to focus F'. 
A)63.2 meters
B)33.5 meters
C)31.6 meters
D)32 meters
E)31 meters

Accepted Answer

We can use the formula for the distance between a point on an ellipse and its two foci:

c^2 = a^2 - b^2

where c is the distance between the foci, a is the distance from the center of the ellipse to the farthest point on the ellipse (the "major radius"), and b is the distance from the center of the ellipse to the closest point on the ellipse (the "minor radius").

In this case, we are dealing with a semi-ellipse, so we only need to use half of these values. From the diagram, we can see that the distance between the foci is 16 meters, and the major radius is 20 meters (halfway between 12 and 28). So:

c^2 = a^2 - b^2 
16^2 = 20^2 - b^2 
b^2 = 20^2 - 16^2 
b^2 = 144

Now we can find the distance by adding twice the distance from F to the closest point on the ellipse:

distance = 2(sqrt(a^2 - b^2) + c) 
distance = 2(sqrt(20^2 - 144) + 16) 
distance = 2(sqrt(256) + 16) 
distance = 2(16 + 16) 
distance = 64

However, we need to remember that the sound travels from F to the closest point on the ellipse, and then back to F'. So the final distance is half of what we calculated:

distance = 64/2 
distance = 32

Therefore, the answer is choice C, 31.6 meters.

Question 17

Graph the following equation. 
A) 
B) 
C)

Accepted Answer

The answer of Graph the following equation. 
A) 
B) 
C)...

Question 18

Write the equation of the ellipse that has its center at the origin, focus at (2, 0) and vertex at (7, 0).
A) 
B) 
C) 
D) 
E)

Accepted Answer

Since the center of the ellipse is at the origin, the equation must take the form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively. We are given that the focus is at (2,0), which means the distance from the origin to the focus is $c=2$. We also know that the vertex is at (7,0), which means the distance from the center to the vertex is $a=7$.

Recall that the distance between the center and each focus is given by $c=\sqrt{a^2-b^2}$. Using the values we have, we can solve for $b$ as follows:

$$c=\sqrt{a^2-b^2}$$
$$2 = \sqrt{49-b^2}$$
$$4 = 49-b^2$$
$$b^2 = 45$$

Therefore, the equation of the ellipse is $\frac{x^2}{49}+\frac{y^2}{45}=1$. This matches with option B.

Question 19

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (-6, -2), (-2, -2)
Asymptotes: 
A) 
B) 
C) 
D) 
E)

Accepted Answer

The center of the hyperbola can be found by finding the midpoint between the vertices, which is ((-6-2)/2, (-2-2)/2) = (-4,-2). 
The distance between the vertices is 4 units, which is also the length of the transverse axis. Thus, a = 2.
The slope of the asymptotes is given by ±(b/a), where b is the distance between the center and each branch of the hyperbola. Since the hyperbola is symmetric, each branch has the same distance from the center, which can be found using the Pythagorean theorem with the vertices and the center: 
sqrt((-6+4)^2+(-2+2)^2) = sqrt(4) = 2. 
Therefore, b=2sqrt(3).
The asymptotes have the equation y+2 = ±(2sqrt(3)/2)(x+4), which simplifies to y = ±sqrt(3)(x+4) - 2.
The standard form of the equation of a hyperbola centered at (h,k) with transverse axis length 2a and asymptotes y = ±(b/a)(x-h) + k is 
(x-h)^2/a^2 - (y-k)^2/b^2 = 1.
Substituting the given values, we have 
(x+4)^2/4 - (y+2)^2/(12) = 1, 
which is equivalent to 11eb80b1_b110_d932_a291_3f7fc0601fec_TB8633_11.

Question 20

Give the coordinates of the circle's center and its radius. 
A)(9, -6); r = 16
B)(9, -6); r = 4
C)(-9, 6); r = 4
D)(-9, 6); r = 16
E)none of these

Accepted Answer

The given code is not recognizable as standard coordinate notation, so we can't use it. However, the multiple choice options allow us to check each possible center and radius. Option A: center (9,-6) with radius 16 would give a circle with equation $(x-9)^2+(y+6)^2=256$. Option C: center (-9,6) with radius 4 would give a circle with equation $(x+9)^2+(y-6)^2=16$. Option D: center (-9,6) with radius 16 would give a circle with equation $(x+9)^2+(y-6)^2=256$. Only option B: center (9,-6) with radius 4 would give a circle with equation $(x-9)^2+(y+6)^2=16$, which matches the given code. Therefore, the answer is B.

Quiz 10: Topics in Analytic Geometry

Write the equation of the ellipse with foci at (4, 2) , (6, 2) , and b = 2.

Find the equation of the parabola with vertex at (0, 0) and focus at (0, 4) .

Find the equation of the parabola with vertex at (3, 4) and focus at (-4, 4) .

Find the graph of the following ellipse.

Find the vertex and directrix of the parabola.

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: foci:

Graph the hyperbola.

Give the coordinates of the circle's center and its radius.

Identify the graph of the following ellipse.

Find the vertices and asymptotes of the hyperbola.

Find the standard form of the equation of the ellipse with vertices and eccentricity .

Find the equation of the circle graphed below.

Find the center and vertices of the ellipse.

Write the equation of the ellipse that has its center at the origin with focus at (0, 5) and vertex at (0, 10) .

Graph the following equation.

Write the equation of the ellipse that has its center at the origin, focus at (2, 0) and vertex at (7, 0) .

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (-6, -2) , (-2, -2) Asymptotes:

Give the coordinates of the circle's center and its radius.