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Mathematics LibreTexts

11.4E: Exercises

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    Practice Makes Perfect

    Exercise \(\PageIndex{15}\) Graph an Ellipse with Center at the Origin

    In the following exercises, graph each ellipse.

    1. \(\frac{x^{2}}{4}+\frac{y^{2}}{25}=1\)
    2. \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
    3. \(\frac{x^{2}}{25}+\frac{y^{2}}{36}=1\)
    4. \(\frac{x^{2}}{16}+\frac{y^{2}}{36}=1\)
    5. \(\frac{x^{2}}{36}+\frac{y^{2}}{16}=1\)
    6. \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\)
    7. \(x^{2}+\frac{y^{2}}{4}=1\)
    8. \(\frac{x^{2}}{9}+y^{2}=1\)
    9. \(4 x^{2}+25 y^{2}=100\)
    10. \(16 x^{2}+9 y^{2}=144\)
    11. \(16 x^{2}+36 y^{2}=576\)
    12. \(9 x^{2}+25 y^{2}=225\)
    Answer

    1.

    This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (2, 0) and (negative 2, 0).
    Figure 11.3.38

    3.

    This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (5, 0) and (negative 5, 0).
    Figure 11.3.39

    5.

    This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).
    Figure 11.3.40

    7.

    This graph shows an ellipse with center (0, 0), vertices (0, 2) and (0, negative 2) and endpoints of minor axis (1, 0) and (negative 1, 0).
    Figure 11.3.41

    9.

    This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).
    Figure 11.3.42

    11.

    This graph shows an ellipse with center (0, 0), vertices (6, 0) and (negative 6, 0) and endpoints of minor axis (0, 4) and (0, negative 4).
    Figure 11.3.43
    Exercise \(\PageIndex{16}\) Find the Equation of an Ellipse with Center at the Origin

    In the following exercises, find the equation of the ellipse shown in the graph.

    1.

    This graph shows an ellipse with center (0, 0), vertices (0, 5) and (0, negative 5) and endpoints of minor axis (negative 3, 0) and (3, 0).
    Figure 11.3.44

    2.

    This graph shows an ellipse with center (0, 0), vertices (5, 0) and (negative 5, 0) and endpoints of minor axis (0, 2) and (0, negative 2).
    Figure 11.3.45

    3.

    This graph shows an ellipse with center (0, 0), vertices (0, 4) and (0, negative 4) and endpoints of minor axis (negative 3, 0) and (3, 0).
    Figure 11.3.46

    4.

    This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 4, 0) and (4, 0).
    Figure 11.3.47
    Answer

    1. \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)

    3. \(\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\)

    Exercise \(\PageIndex{17}\) Graph an Ellipse with Center Not at the Origin

    In the following exercises, graph each ellipse.

    1. \(\frac{(x+1)^{2}}{4}+\frac{(y+6)^{2}}{25}=1\)
    2. \(\frac{(x-3)^{2}}{25}+\frac{(y+2)^{2}}{9}=1\)
    3. \(\frac{(x+4)^{2}}{4}+\frac{(y-2)^{2}}{9}=1\)
    4. \(\frac{(x-4)^{2}}{9}+\frac{(y-1)^{2}}{16}=1\)
    Answer

    1.

    This graph shows an ellipse with center (negative 1, negative 6, vertices (negative 1, negative 1) and (negative 1, negative 11) and endpoints of minor axis (negative 3, negative 6) and (1, negative 6).
    Figure 11.3.48

    3.

    This graph shows an ellipse with center (negative 4, 2, vertices (negative 4, 5) and (negative 4, negative 1) and endpoints of minor axis (3, 1) and (negative 6, 2) and (negative 2, 2).
    Figure 11.3.49
    Exercise \(\PageIndex{18}\) Graph an Ellipse with Center Not at the Origin

    In the following exercises, graph each equation by translation.

    1. \(\frac{(x-3)^{2}}{4}+\frac{(y-7)^{2}}{25}=1\)
    2. \(\frac{(x+6)^{2}}{16}+\frac{(y+5)^{2}}{4}=1\)
    3. \(\frac{(x-5)^{2}}{9}+\frac{(y+4)^{2}}{25}=1\)
    4. \(\frac{(x+5)^{2}}{36}+\frac{(y-3)^{2}}{16}=1\)
    Answer

    1.

    This graph shows an ellipse with center (3, 7), vertices (3, 2) and (3, 12), and endpoints of minor axis (1, 7) and (5, 7).
    Figure 11.3.50

    3.

    This graph shows an ellipse with center (5, negative 4), vertices (5, 1) and (5, negative 9) and endpoints of minor axis (2, negative 4) and (8, negative 4).
    Figure 11.3.51
    Exercise \(\PageIndex{19}\) Graph an Ellipse with Center Not at the Origin

    In the following exercises,

    1. Write the equation in standard form and
    2. Graph.
    1. \(25 x^{2}+9 y^{2}-100 x-54 y-44=0\)
    2. \(4 x^{2}+25 y^{2}+8 x+100 y+4=0\)
    3. \(4 x^{2}+25 y^{2}-24 x-64=0\)
    4. \(9 x^{2}+4 y^{2}+56 y+160=0\)
    Answer

    1.

    1. \(\frac{(x-2)^{2}}{9}+\frac{(y-3)^{2}}{25}=1\)
    This graph shows an ellipse with center (2, 3), vertices (2, negative 2) and (2, 8) and endpoints of minor axis (negative 1, 3) and (5, 3).
    Figure 11.3.52

    3.

    1. \(\frac{y^{2}}{4}+\frac{(x-3)^{2}}{25}=1\)
    This graph shows an ellipse with center (3, 0), vertices (negative 2, 0) and (8, 0) and endpoints of minor axis (3, 2) and (3, negative 2).
    Figure 11.3.53
    Exercise \(\PageIndex{20}\) Graph an Ellipse with Center Not at the Origin

    In the following exercises, graph the equation.

    1. \(x=-2(y-1)^{2}+2\)
    2. \(x^{2}+y^{2}=49\)
    3. \((x+5)^{2}+(y+2)^{2}=4\)
    4. \(y=-x^{2}+8 x-15\)
    5. \(\frac{(x+3)^{2}}{16}+\frac{(y+1)^{2}}{4}=1\)
    6. \((x-2)^{2}+(y-3)^{2}=9\)
    7. \(\frac{x^{2}}{25}+\frac{y^{2}}{36}=1\)
    8. \(x=4(y+1)^{2}-4\)
    9. \(x^{2}+y^{2}=64\)
    10. \(\frac{x^{2}}{9}+\frac{y^{2}}{25}=1\)
    11. \(y=6 x^{2}+2 x-1\)
    12. \(\frac{(x-2)^{2}}{9}+\frac{(y+3)^{2}}{25}=1\)
    Answer

    1.

    This graph shows a parabola with vertex (2, 1) and y intercepts (0, 0) and (2, 0).
    Figure 11.3.54

    3.

    This graph shows a circle with center (negative 5, negative 2) and a radius of 2 units.
    Figure 11.3.55

    5.

    This graph shows an ellipse with center (negative 3, negative 1), vertices (1, negative 1) and (negative 7, negative 1) and endpoints of minor axis (negative 3, 1) and (negative 3, negative 3).
    Figure 11.3.56

    7.

    This graph shows an ellipse with center (0, 0), vertices (0, 6) and (0, negative 6) and endpoints of minor axis (negative 5, 0) and (5, 0).
    Figure 11.3.57

    9.

    This graph shows circle with center (0, 0) and with radius 8 units.
    Figure 11.3.58

    11.

    This graph shows upward opening parabola. Its vertex has an x value of slightly less than 0 and a y value of slightly less than minus 1. A point on it is approximately at (negative 1, 3).
    Figure 11.3.59
    Exercise \(\PageIndex{21}\) Solve Application with Ellipses

    1. A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately \(10\) AU and the furthest is approximately \(30\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.

    This graph shows an ellipse with center (0, 0), vertices (negative 20, 0) and (20, 0). The sun is shown at point (10, 0), which is 30 units from the left vertex and 10 units from the right vertex.
    Figure 11.3.60

    2. A planet moves in an elliptical orbit around its sun. The closest the planet gets to the sun is approximately \(10\) AU and the furthest is approximately \(70\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the planet.

    This graph shows an ellipse with center (0, 0), vertices (negative 40, 0) and (40, 0). The sun is shown at point (30, 0), which is 70 units from the left vertex and 10 units from the right vertex.
    Figure 11.3.61

    3. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(15\) AU and the furthest is approximately \(85\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.

    This graph shows an ellipse with center (0, 0), vertices (negative 50, 0) and (50, 0). The sun is shown at point (35, 0), which is 85 units from the left vertex and 15 units from the right vertex.
    Figure 11.3.62

    4. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(15\) AU and the furthest is approximately \(95\) AU. The sun is one of the foci of the elliptical orbit. Letting the ellipse center at the origin and labeling the axes in AU, the orbit will look like the figure below. Use the graph to write an equation for the elliptical orbit of the comet.

    This graph shows an ellipse with center (0, 0), vertices (negative 55, 0) and (55, 0). The sun is shown at point (40, 0), which is 95 units from the left vertex and 15 units from the right vertex.
    Figure 11.3.63
    Answer

    1. \(\frac{x^{2}}{400}+\frac{y^{2}}{300}=1\)

    3. \(\frac{x^{2}}{2500}+\frac{y^{2}}{1275}=1\)

    Exercise \(\PageIndex{22}\) Writing Exercises
    1. In your own words, define an ellipse and write the equation of an ellipse centered at the origin in standard form. Draw a sketch of the ellipse labeling the center, vertices and major and minor axes.
    2. Explain in your own words how to get the axes from the equation in standard form.
    3. Compare and contrast the graphs of the equations \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\) and \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\).
    4. Explain in your own words, the difference between a vertex and a focus of the ellipse.
    Answer

    1. Answers may vary

    3. Answers may vary

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has 4 columns 4 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first columns has the following statements: graph an ellipse with center at the origin, find the equation of an ellipse with center at the origin, graph an ellipse with center not at the origin, solve applications with ellipses. The remaining columns are blank.
    Figure 11.3.64

    b. What does this checklist tell you about your mastery of this section? What steps will you take to improve?


    11.4E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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