Precalculus : Ellipses

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #2 : Conic Sections

Find the focal points of the conic below:

Possible Answers:

Correct answer:

Explanation:

The first thing we want to do is put the conic (an ellipse because the x2 and the y2 terms have the same sign) into a better form i.e. 

where (h,k) is the center of our ellipse.

We will continue by completing the square for both the x and y binomials.

First we seperate them into two trinomials:

  

then we pull a 27 out of the first one and a 16 out of the second

then we add the correct constant to each trinomial (being sure to add the same amount to the other side of our equation.

then we factor our trinomials and divide by 16 and 27 to get

so the center of our ellipse is (-6,3) and we calculate the distance from the focal points to the center by the equation:

 

and we know that our ellipse is stretched in the y direction because b>a so our focal points will be c displaced from our center. 

with 

our focal points are 

Example Question #1 : Ellipses

Find the center of this ellipse:  

Possible Answers:

Correct answer:

Explanation:

To find the center of this ellipse we need to put it into a better form. We do this by rearranging our terms and completing the square for both our y and x terms.

 

completing the square for both gives us this.

we could divide by 429 but we have the information we need. The center of our ellipse is 

Example Question #3 : Conic Sections

What is the equation of the elipse centered at the origin and passing through the point (5, 0) with major radius 5 and minor radius 3? 

Possible Answers:

Correct answer:

Explanation:

The equation of an ellipse is

,

where a is the horizontal radius, b is the vertical radius, and (h, k) is the center of the ellipse. In this case we are told that the center is at the origin, or (0,0), so both h and k equal 0. That brings us to:

We are told about the major and minor radiuses, but the problem does not specify which one is horizontal and which one vertical. However it does tell us that the ellipse passes through the point (5, 0), which is in a horizontal line with the center, (0, 0). Therefore the horizontal radius is 5.

The vertical radius must then be 3. We can now plug these in:

Example Question #7 : Conic Sections

An ellipse is centered at (-3, 2) and passes through the points (-3, 6) and (4, 2). Determine the equation of this eclipse.

Possible Answers:

Correct answer:

Explanation:

The usual form for an ellipse is

,

where (h, k) is the center of the ellipse, a is the horizontal radius, and b is the vertical radius. 

Plug in the coordinate pair:

Now we have to find the horizontal radius and the vertical radius. Let's compare points; we are told the ellipse passes through the point (-3, 6), which is vertically aligned with the center. Therefore the vertical radius is 4.

Similarly, the ellipse passes through the point (4, 2), which is horizontally aligned with the center. This means the horizontal radius must be 7.

Substitute:

Example Question #8 : Conic Sections

What is the shape of the graph indicated by the equation?

Possible Answers:

Ellipse

Hyperbola

Parabola

Circle

Correct answer:

Ellipse

Explanation:

An ellipse has an equation that can be written in the format. The center is indicated by , or in this case .

Example Question #1 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

Find the endpoints of the major and minor axes of the ellipse described by the following equation:

Possible Answers:

Correct answer:

Explanation:

In order to find the endpoints of the major and minor axes of our ellipse, we must first remember what each part of the equation in standard form means:

The point given by (h,k) is the center of our ellipse, so we know the center of the ellipse in the problem is (8,-2), and we know that the end points of our major and minor axes will line up with the center either in the x or y direction, depending on the axis. The parts of the equation that will tell us the distance from the center to the endpoints of each axis are and . If we take the square root of each, a will give us the distance from the center to the endpoints in the positive and negative x direction, and b will give us the distance from the center to the endpoints in the positive and negative y direction:

Now it is important to consider the definition of major and minor axes. The major axis of an ellipse is the longer one, will the minor axis is the shorter one. We can see that b=5, which means the axis is longer in the y direction, so this is the major axis. To find the endpoints of the major axis, we'll go 5 units from the center in the positive and negative y direction, respectively, giving us:

Similarly, to find the endpoints of the minor axis, we'll go 2 units from the center in the positive and negative x direction, respectively, giving us:

Example Question #2 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

The equation of an ellipse, , is . Which of the following are the correct end points of the MAJOR axis of this ellipse?

Possible Answers:

 and 

 and 

 and 

 and 

 and 

Correct answer:

 and 

Explanation:

First, we must determine if the major axis is a vertical axis or a horizontal axis. We look at our denominators,  and , and see that the larger one is under the -term. Therefore, we know that the greater axis will be a vertical one.

To find out how far the end point are from the center, we simply take . So we know the end points will be  units above and below our center. To find the center, we must remember that for ,

the center will be .

So for our equation, the center will be  units above and below the center give us  and .

Example Question #3 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

Find the endpoints of the major axis for the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is horizontal and the endpoints are  and 

Example Question #4 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

Find the endpoints of the major axis for the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is vertical and the endpoints are  and .

Example Question #1 : Find The Endpoints Of The Major And Minor Axes Of An Ellipse

Find the endpoints of the major axis of the ellipse with the following equation:

Possible Answers:

Correct answer:

Explanation:

Recall the standard form of the equation of an ellipse:

, where  is the center of the ellipse.

When , the major axis is horizontal. In this case,  and  are the endpoints of the major axis.

When  and  are the endpoints of the major axis.

 

For the ellipse in question,  is the center. In addition,  and . Since , the major axis is vertical and the endpoints are  and .

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