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Mathematics LibreTexts

Chapter 11 Review Exercises

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    Chapter Review Exercises

    Distance and Midpoint Formulas; Circles

    Exercise \(\PageIndex{1}\) Use the Distance Formula

    In the following exercises, find the distance between the points. Round to the nearest tenth if needed.

    1. \((-5,1)\) and \((-1,4)\)
    2. \((-2,5)\) and \((1,5)\)
    3. \((8,2)\) and \((-7,-3)\)
    4. \((1,-4)\) and \((5,-5)\)
    Answer

    2. \(d=3\)

    4. \(d=\sqrt{17}, d \approx 4.1\)

    Exercise \(\PageIndex{2}\) Use the Midpoint Formula

    In the following exercises, find the midpoint of the line segments whose endpoints are given.

    1. \((-2,-6)\) and \((-4,-2)\)
    2. \((3,7)\) and \((5,1)\)
    3. \((-8,-10)\) and \((9,5)\)
    4. \((-3,2)\) and \((6,-9)\)
    Answer

    2. \((4,4)\)

    4. \(\left(\frac{3}{2},-\frac{7}{2}\right)\)

    Exercise \(\PageIndex{3}\) Write the Equation of a Circle in Standard Form

    In the following exercises, write the standard form of the equation of the circle with the given information.

    1. radius is \(15\) and center is \((0,0)\)
    2. radius is \(\sqrt{7}\) and center is \((0,0)\)
    3. radius is \(9\) and center is \((-3,5)\)
    4. radius is \(7\) and center is \((-2,-5)\)
    5. center is \((3,6)\) and a point on the circle is \((3,-2)\)
    6. center is \((2,2)\) and a point on the circle is \((4,4)\)
    Answer

    2. \(x^{2}+y^{2}=7\)

    4. \((x+2)^{2}+(y+5)^{2}=49\)

    6. \((x-2)^{2}+(y-2)^{2}=8\)

    Exercise \(\PageIndex{4}\) Graph a Circle

    In the following exercises,

    1. Find the center and radius, then
    2. Graph each circle.
    1. \(2 x^{2}+2 y^{2}=450\)
    2. \(3 x^{2}+3 y^{2}=432\)
    3. \((x+3)^{2}+(y-5)^{2}=81\)
    4. \((x+2)^{2}+(y+5)^{2}=49\)
    5. \(x^{2}+y^{2}-6 x-12 y-19=0\)
    6. \(x^{2}+y^{2}-4 y-60=0\)
    Answer

    2.

    1. radius: \(12,\) center: \((0,0)\)
    The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 0) and the radius of the circle is 12.
    Figure 11.E.1

    4.

    1. radius: \(7,\) center: \((-2,-5)\)
    The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (negative 2, negative 5) and the radius of the circle is 7.
    Figure 11.E.2

    6.

    1. radius: \(8,\) center: \((0,2)\)
    The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The center of the circle is (0, 2) and the radius of the circle is 8.
    Figure 11.E.3

    Parabolas

    Exercise \(\PageIndex{5}\) Graph Vertical Parabolas

    In the following exercises, graph each equation by using its properties.

    1. \(y=x^{2}+4 x-3\)
    2. \(y=2 x^{2}+10 x+7\)
    3. \(y=-6 x^{2}+12 x-1\)
    4. \(y=-x^{2}+10 x\)
    Answer

    2.

    The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 7 to 7. The vertex is (negative five-halves, negative eleven-halves) and the parabola passes through the points (negative 4, negative 1) and (negative 1, negative 1).
    Figure 11.E.4

    4.

    The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 36 to 36. The y-axis of the plane runs from negative 26 to 26. The vertex is (5, 25) and the parabola passes through the points (2, 16) and (8, 16).
    Figure 11.E.5
    Exercise \(\PageIndex{6}\) Graph Vertical Parabolas

    In the following exercises,

    1. Write the equation in standard form, then
    2. Use properties of the standard form to graph the equation.
    1. \(y=x^{2}+4 x+7\)
    2. \(y=2 x^{2}-4 x-2\)
    3. \(y=-3 x^{2}-18 x-29\)
    4. \(y=-x^{2}+12 x-35\)
    Answer

    2.

    1. \(y=2(x-1)^{2}-4\)
    The figure shows an upward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 22 to 22. The y-axis of the plane runs from negative 16 to 16. The vertex is (1, negative 4) and the parabola passes through the points (0, negative 2) and (2, negative 2).
    Figure 11.E.6

    4.

    1. \(y=-(x-6)^{2}+1\)
    The figure shows a downward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 60 to 60. The y-axis of the plane runs from negative 46 to 46. The vertex is (6, 1) and the parabola passes through the points (5, 0) and (7, 0).
    Figure 11.E.7
    Exercise \(\PageIndex{7}\) Graph Horizontal Parabolas

    In the following exercises, graph each equation by using its properties.

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

    2.

    The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (4, negative 1) and the parabola passes through the points (6, 0) and (6, negative 2).
    Figure 11.E.8

    4.

    The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (0, 0) and the parabola passes through the points (negative 3, 1) and (negative 3, negative 1).
    Figure 11.E.9
    Exercise \(\PageIndex{8}\) Graph Horizontal Parabolas

    In the following exercises,

    1. Write the equation in standard form, then
    2. Use properties of the standard form to graph the equation.
    1. \(x=4 y^{2}+8 y\)
    2. \(x=y^{2}+4 y+5\)
    3. \(x=-y^{2}-6 y-7\)
    4. \(x=-2 y^{2}+4 y\)
    Answer

    2.

    1. \(x=(y+2)^{2}+1\)
    The figure shows a rightward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (1, negative 2) and the parabola passes through the points (5, 0) and (5, negative 4).
    Figure 11.E.10

    4.

    1. \(x=-2(y-1)^{2}+2\)
    The figure shows a leftward-opening parabola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The vertex is (2, negative 3) and the parabola passes through the points (0, 2) and (0, 0).
    Figure 11.E.11
    Exercise \(\PageIndex{9}\) Solve Applications with Parabolas

    In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.

    1.

    The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 5 feet high and 20 feet wide.
    Figure 11.E.12

    2.

    The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 25 feet high and 30 feet wide.
    Figure 11.E.13
    Answer

    2. \(y=-\frac{1}{9} x^{2}+\frac{10}{3} x\)

    Ellipses

    Exercise \(\PageIndex{10}\) Graph an Ellipse with Center at the Origin

    In the following exercises, graph each ellipse.

    1. \(\frac{x^{2}}{36}+\frac{y^{2}}{25}=1\)
    2. \(\frac{x^{2}}{4}+\frac{y^{2}}{81}=1\)
    3. \(49 x^{2}+64 y^{2}=3136\)
    4. \(9 x^{2}+y^{2}=9\)
    Answer

    2.

    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9), and co-vertices at (plus or minus 2, 0).
    Figure 11.E.14

    4.

    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 9 to 9. The y-axis of the plane runs from negative 7 to 7. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 3), and co-vertices at (plus or minus 1, 0).
    Figure 11.E.15
    Exercise \(\PageIndex{11}\) 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.

    The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 10, 0), and co-vertices at (0, plus or minus 4).
    Figure 11.E.16

    2.

    The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 8), and co-vertices at (plus or minus 6, 0).
    Figure 11.E.17
    Answer

    2. \(\frac{x^{2}}{36}+\frac{y^{2}}{64}=1\)

    Exercise \(\PageIndex{12}\) Graph an Ellipse with Center Not at the Origin

    In the following exercises, graph each ellipse.

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

    2.

    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 4, negative 1), a horizontal major axis, vertices at (negative 8, negative 1) and (0, negative 1) and co-vertices at (negative 4, 2) and (negative 4, negative 4).
    Figure 11.E.18

    4.

    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (negative 3, 2), a vertical major axis, vertices at (negative 3, 7) and (negative 3, negative 3) and co-vertices at (negative 6, 2) and (0, 2).
    Figure 11.E.19
    Exercise \(\PageIndex{13}\) Graph an Ellipse with Center Not at the Origin

    In the following exercises,

    1. Write the equation in standard form and
    2. Graph.
    1. \(x^{2}+y^{2}+12 x+40 y+120=0\)
    2. \(25 x^{2}+4 y^{2}-150 x-56 y+321=0\)
    3. \(25 x^{2}+4 y^{2}+150 x+125=0\)
    4. \(4 x^{2}+9 y^{2}-126 x+405=0\)
    Answer

    2.

    1. \(\frac{(x-3)^{2}}{4}+\frac{(y-7)^{2}}{25}=1\)
    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 18 to 18. The y-axis of the plane runs from negative 14 to 14. The ellipse has a center at (3, 7), a vertical major axis, vertices at (3, 2) and (3, 12) and co-vertices at (negative 1, 7) and (5, 7).
    Figure 11.E.20

    4.

    1. \(\frac{x^{2}}{9}+\frac{(y-7)^{2}}{4}=1\)
    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 15 to 15. The y-axis of the plane runs from negative 11 to 11. The ellipse has a center at (0, 7), a horizontal major axis, vertices at (3, 7) and (negative 3, 7) and co-vertices at (0, 5) and (0, 9).
    Figure 11.E.21
    Exercise \(\PageIndex{14}\) Solve Applications with Ellipses

    In the following exercises, write the equation of the ellipse described.

    1. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(10\) AU and the furthest is approximately \(90\) 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.
    The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 50, 0), the sun marked as a foci and labeled (50, 0), the closest distance the comet is from the sun marked as 10 A U, and the farthest a comet is from the sun marked as 90 A U.
    Figure 11.E.22
    Answer

    1. Solve

    Hyperbolas

    Exercise \(\PageIndex{15}\) Graph a Hyperbola with Center at \((0,0)\)

    In the following exercises, graph.

    1. \(\frac{x^{2}}{25}-\frac{y^{2}}{9}=1\)
    2. \(\frac{y^{2}}{49}-\frac{x^{2}}{16}=1\)
    3. \(9 y^{2}-16 x^{2}=144\)
    4. \(16 x^{2}-4 y^{2}=64\)
    Answer

    1.

    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 12 to 12. The y-axis of the plane runs from negative 9 to 9. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 5, 0), and that open left and right.
    Figure 11.E.23

    3.

    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 19 to 19. The y-axis of the plane runs from negative 15 to 15. The hyperbola has a center at (0, 0) and branches that pass through the vertices (0, plus or minus 4), and that open up and down.
    Figure 11.E.24
    Exercise \(\PageIndex{16}\) Graph a Hyperbola with Center at \((h,k)\)

    In the following exercises, graph.

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

    1.

    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (1, negative 1), and that open left and right.
    Figure 11.E.25

    3.

    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, negative 2) and branches that pass through the vertices (negative 1, 1) and (negative 1, negative 5), and that open up and down.
    Figure 11.E.26
    Exercise \(\PageIndex{17}\) Graph a Hyperbola with Center at \((h,k)\)

    In the following exercises,

    1. Write the equation in standard form and
    2. Graph.
    1. \(4 x^{2}-16 y^{2}+8 x+96 y-204=0\)
    2. \(16 x^{2}-4 y^{2}-64 x-24 y-36=0\)
    3. \(4 y^{2}-16 x^{2}+32 x-8 y-76=0\)
    4. \(36 y^{2}-16 x^{2}-96 x+216 y-396=0\)
    Answer

    1.

    1. \(\frac{(x+1)^{2}}{16}-\frac{(y-3)^{2}}{4}=1\)
    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (negative 1, 3) and branches that pass through the vertices (negative 5, 3) and (3, 3), and that open left and right.
    Figure 11.E.27

    3.

    1. \(\frac{(y-1)^{2}}{16}-\frac{(x-1)^{2}}{4}=1\)
    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (1, 1) and branches that pass through the vertices (1, negative 3) and (1, 5), and that open up and down.
    Figure 11.E.28
    Exercise \(\PageIndex{18}\) Identify the Graph of Each Equation as a Circle, Parabola, Ellipse, or Hyperbola

    In the following exercises, identify the type of graph.

      1. \(16 y^{2}-9 x^{2}-36 x-96 y-36=0\)
      2. \(x^{2}+y^{2}-4 x+10 y-7=0\)
      3. \(y=x^{2}-2 x+3\)
      4. \(25 x^{2}+9 y^{2}=225\)
      1. \(x^{2}+y^{2}+4 x-10 y+25=0\)
      2. \(y^{2}-x^{2}-4 y+2 x-6=0\)
      3. \(x=-y^{2}-2 y+3\)
      4. \(16 x^{2}+9 y^{2}=144\)
    Answer

    1.

    1. Hyperbola
    2. Circle
    3. Parabola
    4. Ellipse

    Solve Systems of Nonlinear Equations

    Exercise \(\PageIndex{19}\) Solve a System of Nonlinear Equations Using Graphing

    In the following exercises, solve the system of equations by using graphing.

    1. \(\left\{\begin{array}{l}{3 x^{2}-y=0} \\ {y=2 x-1}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{y=x^{2}-4} \\ {y=x-4}\end{array}\right.\)
    3. \(\left\{\begin{array}{l}{x^{2}+y^{2}=169} \\ {x=12}\end{array}\right.\)
    4. \(\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {y=-5}\end{array}\right.\)
    Answer

    1.

    The figure shows a parabola and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 5 to 5. The y-axis of the plane runs from negative 4 to 4. The parabola has a vertex at (0, 0) and opens upward. The line has a slope of 2 with a y-intercept at negative 1. The parabola and line do not intersect, so the system has no solution.
    Figure 11.E.29

    3.

    The figure shows a circle and line graphed on the x y coordinate plane. The x-axis of the plane runs from negative 20 to 20. The y-axis of the plane runs from negative 15 to 15. The circle has a center at (0, 0) and a radius of 13. The line is vertical. The circle and line intersect at the points (12, 5) and (12, negative 5), which are labeled. The solution of the system is (12, 5) and (12, negative 5)
    Figure 11.E.30
    Exercise \(\PageIndex{20}\) Solve a System of Nonlinear Equations Using Substitution

    In the following exercises, solve the system of equations by using substitution.

    1. \(\left\{\begin{array}{l}{y=x^{2}+3} \\ {y=-2 x+2}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x-y=4}\end{array}\right.\)
    3. \(\left\{\begin{array}{l}{9 x^{2}+4 y^{2}=36} \\ {y-x=5}\end{array}\right.\)
    4. \(\left\{\begin{array}{l}{x^{2}+4 y^{2}=4} \\ {2 x-y=1}\end{array}\right.\)
    Answer

    1. \((-1,4)\)

    3. No solution

    Exercise \(\PageIndex{21}\) Solve a System of Nonlinear Equations Using Elimination

    In the following exercises, solve the system of equations by using elimination.

    1. \(\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-2 y-1=0}\end{array}\right.\)
    2. \(\left\{\begin{array}{l}{x^{2}-y^{2}=5} \\ {-2 x^{2}-3 y^{2}=-30}\end{array}\right.\)
    3. \(\left\{\begin{array}{l}{4 x^{2}+9 y^{2}=36} \\ {3 y^{2}-4 x=12}\end{array}\right.\)
    4. \(\left\{\begin{array}{l}{x^{2}+y^{2}=14} \\ {x^{2}-y^{2}=16}\end{array}\right.\)
    Answer

    1. \((-\sqrt{7}, 3),(\sqrt{7}, 3)\)

    3. \((-3,0),(0,-2),(0,2)\)

    Exercise \(\PageIndex{22}\) Use a System of Nonlinear Equations to Solve Applications

    In the following exercises, solve the problem using a system of equations.

    1. The sum of the squares of two numbers is \(25\). The difference of the numbers is \(1\). Find the numbers.
    2. The difference of the squares of two numbers is \(45\). The difference of the square of the first number and twice the square of the second number is \(9\). Find the numbers.
    3. The perimeter of a rectangle is \(58\) meters and its area is \(210\) square meters. Find the length and width of the rectangle.
    4. Colton purchased a larger microwave for his kitchen. The diagonal of the front of the microwave measures \(34\) inches. The front also has an area of \(480\) square inches. What are the length and width of the microwave?
    Answer

    1. \(-3\) and \(-4\) or \(4\) and \(3\)

    3. If the length is \(14\) inches, the width is \(15\) inches. If the length is \(15\) inches, the width is \(14\) inches.

    Practice Test

    Exercise \(\PageIndex{23}\)

    In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.

    1. \((-4,-3)\) and \((-10,-11)\)
    2. \((6,8)\) and \((-5,-3)\)
    Answer

    1. distance: \(10,\) midpoint: \((-7,-7)\)

    Exercise \(\PageIndex{24}\)

    In the following exercises, write the standard form of the equation of the circle with the given information.

    1. radius is \(11\) and center is \((0,0)\)
    2. radius is \(12\) and center is \((10,-2)\)
    3. center is \((-2,3)\) and a point on the circle is \((2,-3)\)
    4. Find the equation of the ellipse shown in the graph.
    The figure shows an ellipse graphed on the x y coordinate plane. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 10), and co-vertices at (plus or minus 6, 0).
    Figure 11.E.31
    Answer

    1. \(x^{2}+y^{2}=121\)

    3. \((x+2)^{2}+(y-3)^{2}=52\)

    Exercise \(\PageIndex{25}\)

    In the following exercises,

    1. Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and
    2. Graph the equation.
    1. \(4 x^{2}+49 y^{2}=196\)
    2. \(y=3(x-2)^{2}-2\)
    3. \(3 x^{2}+3 y^{2}=27\)
    4. \(\frac{y^{2}}{100}-\frac{x^{2}}{36}=1\)
    5. \(\frac{x^{2}}{16}+\frac{y^{2}}{81}=1\)
    6. \(x=2 y^{2}+10 y+7\)
    7. \(64 x^{2}-9 y^{2}=576\)
    Answer

    1.

    1. Ellipse
    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The ellipse has a center at (0, 0), a horizontal major axis, vertices at (plus or minus 7, 0) and co-vertices at (0, plus or minus 2).
    Figure 11.E.32

    3.

    1. Circle
    The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The parabola circle has a center at (0, 0) and a radius of 3.
    Figure 11.E.33

    5.

    1. Ellipse
    The figure shows an ellipse graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The ellipse has a center at (0, 0), a vertical major axis, vertices at (0, plus or minus 9) and co-vertices at (plus or minus 4, 0).
    Figure 11.E.34

    7.

    1. Hyperbola
    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 8 to 8. The hyperbola has a center at (0, 0) and branches that pass through the vertices (plus or minus 3, 0) and that open left and right.
    Figure 11.E.35
    Exercise \(\PageIndex{26}\)

    In the following exercises,

    1. Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola,
    2. Write the equation in standard form, and
    3. Graph the equation.
    1. \(25 x^{2}+64 y^{2}+200 x-256 y-944=0\)
    2. \(x^{2}+y^{2}+10 x+6 y+30=0\)
    3. \(x=-y^{2}+2 y-4\)
    4. \(9 x^{2}-25 y^{2}-36 x-50 y-214=0\)
    5. \(y=x^{2}+6 x+8\)
    6. Solve the nonlinear system of equations by graphing: \(\left\{\begin{array}{l}{3 y^{2}-x=0} \\ {y=-2 x-1}\end{array}\right.\).
    7. Solve the nonlinear system of equations using substitution: \(\left\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {y=-x-4}\end{array}\right.\).
    8. Solve the nonlinear system of equations using elimination: \(\left\{\begin{array}{l}{x^{2}+9 y^{2}=9} \\ {2 x^{2}-9 y^{2}=18}\end{array}\right.\)
    9. Create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in \(y=a x^{2}+b x+c\) form.
    The figure shows a parabolic arch formed in the foundation of the bridge. The arch is 10 feet high and 30 feet wide.
    Figure 11.E.36

    10. A comet moves in an elliptical orbit around a sun. The closest the comet gets to the sun is approximately \(20\) 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 comet.

    The figure shows a model of an elliptical orbit around the sun on the x y coordinate plane. The ellipse has a center at (0, 0), a horizontal major axis, vertices marked at (plus or minus 45, 0), the sun marked as a foci and labeled (25, 0), the closest distance the comet is from the sun marked as 20 A U, and the farthest a comet is from the sun marked as 70 A U.
    Figure 11.E.37

    11. The sum of two numbers is \(22\) and the product is \(−240\). Find the numbers.

    12. For her birthday, Olive’s grandparents bought her a new widescreen TV. Before opening it she wants to make sure it will fit her entertainment center. The TV is \(55\)”. The size of a TV is measured on the diagonal of the screen and a widescreen has a length that is larger than the width. The screen also has an area of \(1452\) square inches. Her entertainment center has an insert for the TV with a length of \(50\) inches and width of \(40\) inches. What are the length and width of the TV screen and will it fit Olive’s entertainment center?

    Answer

    2.

    1. Circle
    2. \((x+5)^{2}+(y+3)^{2}=4\)
    The figure shows a circle graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The circle has a center at (negative 5, negative 3) and a radius 2.
    Figure 11.E.38

    4.

    1. Hyperbola
    2. \(\frac{(x-2)^{2}}{25}-\frac{(y+1)^{2}}{9}=1\)
    The figure shows a hyperbola graphed on the x y coordinate plane. The x-axis of the plane runs from negative 14 to 14. The y-axis of the plane runs from negative 10 to 10. The hyperbola has a center at (2, negative 1) and branches that pass through the vertices (negative 3, negative 1) and (7, negative 1) that open left and right.
    Figure 11.E.39

    6. No solution

    8. \((0,-3),(0,3)\)

    10. \(\frac{x^{2}}{2025}+\frac{y^{2}}{1400}=1\)

    12. The length is \(44\) inches and the width is \(33\) inches. The TV will fit Olive’s entertainment center.

    Glossary

    system of nonlinear equations
    A system of nonlinear equations is a system where at least one of the equations is not linear.

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