Question 1

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

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Question 2

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

Accepted Answer

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A) 
B) 
C) 
D)...

Question 3

Write the equation of the ellipse that has its center at the origin with focus at (0, 2) and vertex at (0, 10).
A) 
B) 
C) 
D) 
E)none of these

Accepted Answer

Since the center is at the origin, we have $h = 0$ and $k = 0$.

Since the distance from the center to the focus is 2, we have $c = 2$.

Since the distance from the center to the vertex is 10, we have $a = 10$.

Recall that the standard form of the equation of an ellipse with a horizontal major axis is:

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

where $a$ is the distance from the center to the vertex (in our case, 10), and $b$ is the distance from the center to the co-vertex. We know that $c^2 = a^2 - b^2$, so:

$$2^2 = 10^2 - b^2 \quad\Rightarrow\quad b^2 = 96$$

Therefore, the equation of the ellipse is:

$$\frac{x^2}{10^2} + \frac{y^2}{96} = 1$$

Multiplying both sides by 96, we get:

$$\frac{96x^2}{100} + y^2 = 96$$

Simplifying and rearranging, we get:

$$\frac{x^2}{\frac{100}{96}} + \frac{y^2}{96} = 1$$

So the final equation is:

$$\frac{x^2}{\frac{25}{24}} + \frac{y^2}{96} = 1$$

which matches the equation in choice B.

Question 4

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 5

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

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Question 6

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

Accepted Answer

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A) 
B) 
C) 
D)...

Question 7

In a suspension bridge, the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 1000 m apart, and the lowest point of the suspension cables is 200 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 = -1,250y B)x² = 1,250y C)x² = 5,000y D)x² = 2,500y E)y² = 5,000x

Accepted Answer

The shape of the suspension cables is parabolic, and we are told that the lowest point of the cable is 200 m below the top of the towers. This means that the vertex of the parabola is located 200 meters below the top of the towers, which is also the origin of our coordinate system.

We are also given that the towers are 1000 meters apart, so the x-coordinate of the two points where the parabolic cables connect to the towers will be -500 and 500.

Let's call the y-coordinate of the vertex "h". Since the parabola is symmetric around its vertex, the y-coordinate of the two points where the parabolic cables connect to the towers will also be "h".

Using the standard form equation for a parabola, y = a(x-h)^2 + k, we know that the vertex is (0, -200) which gives us:

-200 = a(0 - 0)^2 + h

h = -200

Now we need to find the value of "a". To do this, we can use one of the points where the parabolic cables connect to a tower. Let's use the point (-500, -200) since it has the smallest absolute value for the x-coordinate.

-200 = a(-500 - 0)^2 - 200

-200 = 250000a

a = -0.0008

Therefore, our equation for the parabolic part of the cables is:

y = -0.0008(x^2) - 200

or

x^2 = -1250y

Question 8

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

Accepted Answer

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Question 9

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

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Question 10

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (0, 3), (8, 3)
Asymptotes: 
A) 
B) 
C) 
D) 
E)

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Question 11

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)28.3 meters
B)28.6 meters
C)29.1 meters
D)30.1 meters
E)57.2 meters

Accepted Answer

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Question 12

Write the equation of the ellipse with foci at (2, 8), (8, 8), and b = 2.
A) 
B) 
C) 
D) 
E)

Accepted Answer

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

Question 13

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 14

Find the standard form of the equation of the ellipse with the given characteristics. foci:  endpoints of the major axis: 
A) 
B) 
C) 
D) 
E)

Accepted Answer

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Question 15

Find the equation of the parabola with vertex at (4, 5) and focus at (-3, 5).
A) 
B) 
C) 
D) 
E)

Accepted Answer

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Question 16

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

Accepted Answer

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

Question 17

Identify the conic by writing the equation in standard form. 
A)  ; circle
B)  ; ellipse
C)  ; ellipse
D)  ; ellipse
E)  ; ellipse

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Question 18

Write the equation of the ellipse that has its center at the origin, focus at (4, 0) and vertex at (7, 0).
A) 
B) 
C) 
D) 
E)

Accepted Answer

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Question 19

Find the equation of the hyperbola with vertices (4, 0), (-4, 0) and focus (6, 0).
A) 
B) 
C) 
D) 
E)none of these0

Accepted Answer

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Question 20

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

Accepted Answer

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Quiz 10: Topics in Analytic Geometry

Find the center and vertices of the ellipse.

Graph the hyperbola.

Write the equation of the ellipse that has its center at the origin with focus at (0, 2) and vertex at (0, 10) .

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

Find the vertices and asymptotes of the hyperbola.

Graph the hyperbola.

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

Find the graph of the following ellipse.

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (0, 3) , (8, 3) Asymptotes:

Write the equation of the ellipse with foci at (2, 8) , (8, 8) , and b = 2.

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

Find the standard form of the equation of the ellipse with the given characteristics. foci: endpoints of the major axis:

Find the equation of the parabola with vertex at (4, 5) and focus at (-3, 5) .

Graph the hyperbola.

Identify the conic by writing the equation in standard form.

Write the equation of the ellipse that has its center at the origin, focus at (4, 0) and vertex at (7, 0) .

Find the equation of the hyperbola with vertices (4, 0) , (-4, 0) and focus (6, 0) .

Identify the graph of the following ellipse.