College Algebra : College Algebra

Study concepts, example questions & explanations for College Algebra

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Example Questions

Example Question #121 : College Algebra

The graph of the equation 

is an example of which conic section?

Possible Answers:

A horizontal ellipse

A horizontal hyperbola 

The equation has no graph.

A vertical ellipse

A vertical hyperbola 

Correct answer:

A vertical ellipse

Explanation:

The quadratic coefficients in this general form of a conic equation are 16 and 12. They are of the same sign, making its graph, if it exists, an ellipse. 

To determine whether this ellipse is horizontal or vertical, rewrite this equation in standard form

as follows:

Separate the  and  terms:

Distribute out the quadratic coefficients:

Complete the square within each quadratic expression by dividing each linear coefficient by 2 and squaring the quotient.

Since  and , we get

Balance this equation, adjusting for the distributed coefficients:

The perfect square trinomials are squares of binomials, by design; rewrite them as such:

Divide by 192:

The ellipse is now in standard form. , so the graph of the equation is a vertical ellipse 

Example Question #4 : Ellipses

Give the foci of the ellipse of the equation

Round your coordinates to the nearest tenth, if applicable.

Possible Answers:

 and 

None of the other choices gives the correct response.

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

The equation of the ellipse is given in the standard form

This ellipse has its center at the origin . Also, since , it follows that the ellipse is horizontal. The foci are therefore along the horizontal axis of the ellipse; their coordinates are , where

Substituting 46 and 19 for  and , respectively,

.

The foci are at the points  and .  

Example Question #5 : Ellipses

Give the eccentricity of the ellipse of the equation 

Possible Answers:

Correct answer:

Explanation:

This ellipse is in standard form

where . This is a vertical ellipse, whose foci are 

units from its center in a vertical direction.

The eccentricity of this  ellipse can be calculated by taking the ratio , or, equivalently, . Set  - making  - and . The eccentricity is calculated to be

.

Example Question #11 : Ellipses

The graph of the equation 

is an example of which conic section?

Possible Answers:

A vertical ellipse

The equation has no graph.

A vertical hyperbola 

A horizontal ellipse

A horizontal hyperbola 

Correct answer:

The equation has no graph.

Explanation:

The quadratic coefficients in this general form of a conic equation are 16 and 12. They are of the same sign, making its graph, if it exists, an ellipse. 

To determine whether this ellipse is horizontal or vertical, rewrite this equation in standard form

as follows:

Subtract 384 from both sides:

Separate the  and  terms and group them:

Distribute out the quadratic coefficients:

Complete the square within each quadratic expression by dividing each linear coefficient by 2 and squaring the quotient.

Since  and , we get

Balance this equation, adjusting for the distributed coefficients:

The perfect square trinomials are squares of binomials, by design; rewrite them as such:

Divide by :

Recall that the standard form of an ellipse is

This requires both denominators to be positive. In the standard form of the given equation, they are not.  Therefore, the equation has no real ordered pairs as solutions, and it does not have a graph on the coordinate plane. 

Example Question #121 : College Algebra

Give the foci of the ellipse of the equation

.

Round your coordinates to the nearest tenth, if applicable.

Possible Answers:

None of the other choices gives the correct response.

Correct answer:

Explanation:

is the standard form of an ellipse with center . Also, since in the given equation, and - that is, , the ellipse is horizontal.

The foci of a horizontal ellipse are located at

,

where

Setting , the foci are at

, or

and .

Example Question #2 : Conic Sections

Using the information below, determine the equation of the hyperbola.

Foci:  and 

Eccentricity: 

Possible Answers:

Correct answer:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci = 

Distance between vertices = 

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 12, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of 

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 4, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 12, we can conclude that 

Center point: 

Thus, the equation of the hyperbola is:

Example Question #11 : Conic Sections

Using the information below, determine the equation of the hyperbola.

Foci:  and 

Eccentricity: 

Possible Answers:

Correct answer:

Explanation:

General Information for Hyperbola:

Equation for horizontal transverse hyperbola:

Distance between foci = 

Distance between vertices = 

Eccentricity =

Center: (h, k)

First determine the value of c. Since we know the distance between the two foci is 8, we can set that equal to .

Next, use the eccentricity equation and the value of the eccentricity provided in the question to determine the value of a.

Eccentricity =

Determine the value of 

Determine the center point to identify the values of h and k. Since the y coordinate of the foci are 8, the center point will be on the same line. Hence, .

Since center point is equal distance from both foci, and we know that the distance between the foci is 8, we can conclude that 

Center point: 

Thus, the equation of the hyperbola is:

Example Question #1 : Hyperbolas

Find the foci of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

 and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at  and .

Example Question #2 : Hyperbolas

Find the foci of a hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

 and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at  and .

Example Question #111 : Understand Features Of Hyperbolas And Ellipses

Find the foci of the hyperbola with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall that the standard formula of a hyperbola can come in two forms:

 and

, where the centers of both hyperbolas are .

When the term with  is first, that means the foci will lie on a horizontal transverse axis.

When the term with  is first, that means the foci will lie on a vertical transverse axis.

To find the foci, we use the following:

For a hyperbola with a horizontal transverse access, the foci will be located at  and .

For a hyperbola with a vertical transverse access, the foci will be located at  and .

For the given hypebola in the question, the transverse axis is horizontal and its center is located at .

Next, find .

The foci are then located at  and .

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