Question 1

Match the equation of a hyperbola with its description.
-$$\frac { ( y - 3 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 81 } = 1$$
A) center (-1, -3); horizontal transverse axis
B) center (-3, -1); horizontal transverse axis
C) center (3, 1); vertical transverse axis
D) center (1, 3); vertical transverse axis

Accepted Answer

The equation of a hyperbola in the form $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ has its center at (h, k) and a vertical transverse axis if the $y$-term is positive. Here, the center is (1, 3), indicating a vertical transverse axis.

Question 2

Provide an appropriate response.
-If an ellipse has vertices at $( - 2,0 ) , ( 2,0 ) , ( 0,1 )$, and $( 0 , - 1 )$, what is its domain? What is its range?
A) Domain: $[ - 2,2 ]$; Range: $[ - 1,1 ]$
B) Domain: [0, 2]; Range: $[ 0,1 ]$
C) Domain: $[ - 2,1 ] ;$ Range: $[ - 1,2 ]$
D) Domain: $[ - 1,1 ] ;$ Range: $[ - 2,2 ]$

Accepted Answer

The vertices of the ellipse indicate its maximum extents in the x and y directions. The x-coordinates of the vertices are -2 and 2, which define the domain, and the y-coordinates of the vertices are -1 and 1, which define the range.

Question 3

Provide an appropriate response.
-You have to draw an ellipse such that the length of the major axis is fixed at 20 cm. The 
length of the minor axis can have the values 10, 14, or 16 cm. You are told that your ellipse
should be as flat as possible. Which value of length of minor axis would you choose and
why? (Do not calculate).

Accepted Answer

The answer of Provide an appropriate response.
-You have to draw...

Question 4

Provide an appropriate response.
-The eccentricity of ellipse $\mathrm { A }$ is $\mathrm { e } _ { \mathrm { A } } = 0.18$ and the eccentricity of ellipse $\mathrm { B }$ is $\mathrm { eB } = 0.82$. Which ellipse is flatter?

Accepted Answer

The answer of Provide an appropriate response.
-The eccentricity of ellipse...

Question 5

Provide an appropriate response.
-It can be shown that an ellipse with foci $( \mathrm { c } , 0 )$ and $( - \mathrm { c } , 0 )$ where the sum of the distances from any point $( x , y )$ of the ellipse to the two foci is $2 a$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { a ^ { 2 } - c ^ { 2 } } = 1$. Use this result to find an equation of an ellipse with foci $( 3,0 )$ and $( - 3,0 )$, where the sum of the distances from any point of the ellipse to the two foci is 8 .
A) $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 13 } = 1$
B) $x ^ { 2 } + y ^ { 2 } = 64$
C) $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 64 } = 1$
D) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$

Accepted Answer

The given ellipse has foci at $(3,0)$ and $(-3,0)$, so $c=3$. The sum of distances to the foci is $2a=8$, so $a=4$. The equation of the ellipse is $\frac{x^2}{a^2} + \frac{y^2}{a^2-c^2} = 1$. Substituting $a=4$ and $c=3$ gives $\frac{x^2}{16} + \frac{y^2}{16-9} = 1$, which simplifies to $\frac{x^2}{16} + \frac{y^2}{7} = 1$.

Question 6

Choose the equation that matches the graph.
-
A) $\frac { x ^ { 2 } } { 49 } - \frac { y ^ { 2 } } { 16 } = 1$

B) $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 16 } = 1$

C) $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 49 } = 1$

D) $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 49 } = 1$

Accepted Answer

The answer of Choose the equation that matches the graph.
-
A)...

Question 7

Find the eccentricity e of the ellipse.
-A satellite is to be put into an elliptical orbit around a moon. The moon is a sphere with radius $606 \mathrm {~km}$. Determine an equation for the ellipse if the distance of the satellite from the surface of moon varies from $308 \mathrm {~km}$ to $988 \mathrm {~km}$.

A) $\frac { x ^ { 2 } } { 1594 ^ { 2 } } + \frac { y ^ { 2 } } { 914 ^ { 2 } } = 1$

B) $\frac { x ^ { 2 } } { 988 } + \frac { y ^ { 2 } } { 308 } = 1$

C) $\frac { x ^ { 2 } } { 914 ^ { 2 } } + \frac { y ^ { 2 } } { 1594 ^ { 2 } } = 1$

D) $\frac { x ^ { 2 } } { 308 } + \frac { y ^ { 2 } } { 988 } = 1$

Accepted Answer

The answer of Find the eccentricity e of the ellipse.
-A...

Question 8

Find the eccentricity e of the ellipse.
-A rectangular board is 8 units by 12 units. How far from the long side of the board will the foci be located to determine the largest elliptical tabletop?
A) 2.0 units
B) 4.5 units
C) 1.5 units
D) 4.0 units

Accepted Answer

The semi-major axis (a) is 6 units and the semi-minor axis (b) is 4 units. The eccentricity e is calculated using the formula e = sqrt(1 - (b^2/a^2)) = sqrt(1 - (16/36)) = sqrt(20/36) = sqrt(5/9) ≈ 0.745. The distance from the center to each focus is ae = 6 * 0.745 ≈ 4.47 units, which is approximately 1.5 units from the long side (6 - 4.47 = 1.53).

Question 9

Find the eccentricity e of the ellipse.
-A rectangular board is 6 feet by 20 feet. How far from the center of the board will the foci be located to determine the largest elliptical tabletop? Round your answer to the nearest tenth.
A) 7.0 ft
B) 0.5 ft
C) 9.5 ft
D) 3.0 ft

Accepted Answer

The answer of Find the eccentricity e of the ellipse.
-A...

Question 10

Provide an appropriate response.
-An ellipse has eccentricity between 0 and 1. 114)

Accepted Answer

An ellipse is defined by its eccentricity, which is a measure of how much it deviates from being circular. The eccentricity of an ellipse is always between 0 and 1, where 0 represents a perfect circle and values approaching 1 indicate an increasingly elongated shape.

Question 11

Provide an appropriate response.
-$$\frac { x ^ { 2 } } { 5476 } + \frac { y ^ { 2 } } { 5184 } = 1$$ where x and y are measured in millions of miles. Find the eccentricity of this ellipse. (Round to the nearest thousandth.)
A) 1.056
B) 0.231
C) 0.947
D) 3.946

Accepted Answer

The eccentricity of an ellipse is given by $$e = \sqrt{1 - \frac{b^2}{a^2}}$$, where $a$ and $b$ are the semi-major and semi-minor axes, respectively. In the given equation, $\frac{x^2}{5476} + \frac{y^2}{5184} = 1$, we can identify $a^2 = 5476$ and $b^2 = 5184$. Thus, $$e = \sqrt{1 - \frac{5184}{5476}} = \sqrt{1 - \frac{5184}{5476}} = \sqrt{1 - 0.946} = \sqrt{0.054} \approx 0.231$$.

Question 12

Choose the equation that matches the graph.
-
A) $\frac { ( y - 2 ) ^ { 2 } } { 36 } - \frac { ( x + 4 ) ^ { 2 } } { 9 } = 1$

B) $\frac { ( x + 4 ) ^ { 2 } } { 9 } + \frac { ( y - 2 ) ^ { 2 } } { 36 } = 1$

C) $\frac { ( x + 4 ) ^ { 2 } } { 36 } + \frac { ( y - 2 ) ^ { 2 } } { 9 } = 1$

D) $\frac { ( x + 4 ) ^ { 2 } } { 9 } - \frac { ( y - 2 ) ^ { 2 } } { 36 } = 1$

Accepted Answer

The answer of Choose the equation that matches the graph.
-
A)...

Question 13

Find the eccentricity e of the ellipse.
-A rectangular board is 19 by 20 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the
Length of the string.
A) 20 units
B) 38 units
C) 19 units
D) 40 units

Accepted Answer

The answer of Find the eccentricity e of the ellipse.
-A...

Question 14

Find the eccentricity e of the ellipse.
-A railroad tunnel is shaped like a semi-ellipse. The height of the tunnel at the center is $30 \mathrm { ft }$ and the vertical clearance must be $20 \mathrm { ft }$ at a point $5 \mathrm { ft }$ from the center. Find an equation for the ellipse.

A) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 900 } = 1$
B) $\frac { x ^ { 2 } } { 45 } + \frac { y ^ { 2 } } { 400 } = 1$
C) $\frac { x ^ { 2 } } { 900 } + \frac { y ^ { 2 } } { 45 } = 1$
D) $\frac { x ^ { 2 } } { 45 } + \frac { y ^ { 2 } } { 900 } = 1$

Accepted Answer

The answer of Find the eccentricity e of the ellipse.
-A...

Question 15

Choose the equation that matches the graph.
-
A) $\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 36 } = 1$

B) $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 36 } = 1$

C) $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 4 } = 1$

D) $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 4 } = 1$

Accepted Answer

The answer of Choose the equation that matches the graph.
-
A)...

Question 16

Provide an appropriate response.
-$$\text { Discuss the graph of the equation } \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = b \text {. }$$

Accepted Answer

The answer of Provide an appropriate response.
-\[\text { Discuss the...

Question 17

Provide an appropriate response.
-Discuss the symmetries exhibited by an ellipse centered at the origin.

Accepted Answer

The answer of Provide an appropriate response.
-Discuss the symmetries exhibited...

Question 18

Provide an appropriate response.
-In the ellipse $\frac { ( y + 8 ) ^ { 2 } } { 36 } + \frac { ( x - 9 ) ^ { 2 } } { 16 } = 1$, what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?
A) Horizontal: $y = 8$; Vertical: $x = - 9$
B) Horizontal: $y = 9$; Vertical: $x = - 8$
C) Horizontal: $y = 8$; Vertical: $x = 9$
D) Horizontal: $y = - 8$; Vertical: $x = 9$

Accepted Answer

The axes of symmetry for an ellipse are determined by its center, which is at the point $(h, k)$ in the general equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. For the given ellipse, the center is at $(9, -8)$, making the horizontal axis of symmetry $y = -8$ and the vertical axis of symmetry $x = 9$.

Question 19

Match the equation of a hyperbola with its description.
-$$\frac { ( x + 2 ) ^ { 2 } } { 25 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$$
A) center (3, 2); vertical transverse axis
B) center (-2, -3); horizontal transverse axis
C) center (2, 3); horizontal transverse axis
D) center (-3, -2); vertical transverse axis

Accepted Answer

The equation of a hyperbola in the form $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ has a center at (h, k) and a horizontal transverse axis if $a^2$ is under the $x$ term. For the given equation, the center is at (-2, -3), and since the $x$ term is positive and comes first, it has a horizontal transverse axis, matching option B.

Question 20

Find the eccentricity e of the ellipse.
-An elliptical riding path is to be built on a rectangular piece of property that measures 10 mi by 6 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4
Sides 
A) $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 6 } = 1$
B) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 100 } = 1$
C) $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$
D) $\frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 9 } = 1$

Accepted Answer

The answer of Find the eccentricity e of the ellipse.
-An...

Quiz 11: Analytic Geometry

Match the equation of a hyperbola with its description. - $\frac { ( y - 3 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 81 } = 1$

Provide an appropriate response. -If an ellipse has vertices at $( - 2,0 ) , ( 2,0 ) , ( 0,1 )$ , and $( 0 , - 1 )$ , what is its domain? What is its range?

Provide an appropriate response. -The eccentricity of ellipse $\mathrm { A }$ is $\mathrm { e } _ { \mathrm { A } } = 0.18$ and the eccentricity of ellipse $\mathrm { B }$ is $\mathrm { eB } = 0.82$ . Which ellipse is flatter?

Find the eccentricity e of the ellipse. -A rectangular board is 8 units by 12 units. How far from the long side of the board will the foci be located to determine the largest elliptical tabletop?

Find the eccentricity e of the ellipse. -A rectangular board is 6 feet by 20 feet. How far from the center of the board will the foci be located to determine the largest elliptical tabletop? Round your answer to the nearest tenth.

Provide an appropriate response. -An ellipse has eccentricity between 0 and 1. 114)

Provide an appropriate response. - $\frac { x ^ { 2 } } { 5476 } + \frac { y ^ { 2 } } { 5184 } = 1$ where x and y are measured in millions of miles. Find the eccentricity of this ellipse. (Round to the nearest thousandth.)

Find the eccentricity e of the ellipse. -A rectangular board is 19 by 20 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the Length of the string.

Provide an appropriate response. - $\text { Discuss the graph of the equation } \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = b \text {. }$

Provide an appropriate response. -Discuss the symmetries exhibited by an ellipse centered at the origin.

Provide an appropriate response. -In the ellipse $\frac { ( y + 8 ) ^ { 2 } } { 36 } + \frac { ( x - 9 ) ^ { 2 } } { 16 } = 1$ , what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?

Match the equation of a hyperbola with its description. - $\frac { ( x + 2 ) ^ { 2 } } { 25 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$

Quiz 11: Analytic Geometry

Match the equation of a hyperbola with its description. - (y−3) 216−(x−1) 281=1\frac { ( y - 3 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 81 } = 116(y−3) 2​−81(x−1) 2​=1

Provide an appropriate response. -If an ellipse has vertices at (−2,0) ,(2,0) ,(0,1) ( - 2,0 ) , ( 2,0 ) , ( 0,1 ) (−2,0) ,(2,0) ,(0,1) , and (0,−1) ( 0 , - 1 ) (0,−1) , what is its domain? What is its range?

Provide an appropriate response. -The eccentricity of ellipse A\mathrm { A }A is eA=0.18\mathrm { e } _ { \mathrm { A } } = 0.18eA​=0.18 and the eccentricity of ellipse B\mathrm { B }B is eB=0.82\mathrm { eB } = 0.82eB=0.82 . Which ellipse is flatter?

Choose the equation that matches the graph. -

Find the eccentricity e of the ellipse. -A rectangular board is 8 units by 12 units. How far from the long side of the board will the foci be located to determine the largest elliptical tabletop?

Find the eccentricity e of the ellipse. -A rectangular board is 6 feet by 20 feet. How far from the center of the board will the foci be located to determine the largest elliptical tabletop? Round your answer to the nearest tenth.

Provide an appropriate response. -An ellipse has eccentricity between 0 and 1. 114)

Provide an appropriate response. - x25476+y25184=1\frac { x ^ { 2 } } { 5476 } + \frac { y ^ { 2 } } { 5184 } = 15476x2​+5184y2​=1 where x and y are measured in millions of miles. Find the eccentricity of this ellipse. (Round to the nearest thousandth.)

Choose the equation that matches the graph. -

Find the eccentricity e of the ellipse. -A rectangular board is 19 by 20 units. The foci of an ellipse are located to produce the largest area. A string is connected to the foci and pulled taut by a pencil in order to draw the ellipse. Find the Length of the string.

Choose the equation that matches the graph. -

Provide an appropriate response. - Discuss the graph of the equation x2a2+y2b2=1 if a=b. \text { Discuss the graph of the equation } \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = b \text {. } Discuss the graph of the equation a2x2​+b2y2​=1 if a=b.

Provide an appropriate response. -Discuss the symmetries exhibited by an ellipse centered at the origin.

Provide an appropriate response. -In the ellipse (y+8) 236+(x−9) 216=1\frac { ( y + 8 ) ^ { 2 } } { 36 } + \frac { ( x - 9 ) ^ { 2 } } { 16 } = 136(y+8) 2​+16(x−9) 2​=1 , what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?

Match the equation of a hyperbola with its description. - (x+2) 225−(y+3) 216=1\frac { ( x + 2 ) ^ { 2 } } { 25 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 125(x+2) 2​−16(y+3) 2​=1

Find the eccentricity e of the ellipse. -An elliptical riding path is to be built on a rectangular piece of property that measures 10 mi by 6 mi. Find an equation for the ellipse if the path is to touch the center of the property line on all 4 Sides

Match the equation of a hyperbola with its description. - $\frac { ( y - 3 ) ^ { 2 } } { 16 } - \frac { ( x - 1 ) ^ { 2 } } { 81 } = 1$

Provide an appropriate response. -If an ellipse has vertices at $( - 2,0 ) , ( 2,0 ) , ( 0,1 )$ , and $( 0 , - 1 )$ , what is its domain? What is its range?

Provide an appropriate response. -The eccentricity of ellipse $\mathrm { A }$ is $\mathrm { e } _ { \mathrm { A } } = 0.18$ and the eccentricity of ellipse $\mathrm { B }$ is $\mathrm { eB } = 0.82$ . Which ellipse is flatter?

Provide an appropriate response. - $\frac { x ^ { 2 } } { 5476 } + \frac { y ^ { 2 } } { 5184 } = 1$ where x and y are measured in millions of miles. Find the eccentricity of this ellipse. (Round to the nearest thousandth.)

Provide an appropriate response. - $\text { Discuss the graph of the equation } \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \text { if } a = b \text {. }$

Provide an appropriate response. -In the ellipse $\frac { ( y + 8 ) ^ { 2 } } { 36 } + \frac { ( x - 9 ) ^ { 2 } } { 16 } = 1$ , what is the equation of the horizontal axis of symmetry? The vertical axis of symmetry?

Match the equation of a hyperbola with its description. - $\frac { ( x + 2 ) ^ { 2 } } { 25 } - \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$