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Example Questions
Example Question #121 : College Algebra
The graph of the equation
is an example of which conic section?
A horizontal ellipse
A horizontal hyperbola
The equation has no graph.
A vertical ellipse
A vertical hyperbola
A vertical ellipse
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.
and
None of the other choices gives the correct response.
and
and
and
and
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
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?
A vertical ellipse
The equation has no graph.
A vertical hyperbola
A horizontal ellipse
A horizontal hyperbola
The equation has no graph.
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.
None of the other choices gives the correct response.
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:
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:
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:
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:
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:
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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