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Chapter 11 Review Exercises

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Chapter Review Exercises

Distance and Midpoint Formulas; Circles

Exercise 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=√17,d≈4.1

Exercise 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. (32,−72)

Exercise 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 √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. x2+y2=7

4. (x+2)2+(y+5)2=49

6. (x−2)2+(y−2)2=8

Exercise 4 Graph a Circle

In the following exercises,

  1. Find the center and radius, then
  2. Graph each circle.
  1. 2x2+2y2=450
  2. 3x2+3y2=432
  3. (x+3)2+(y−5)2=81
  4. (x+2)2+(y+5)2=49
  5. x2+y2−6x−12y−19=0
  6. x2+y2−4y−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 5 Graph Vertical Parabolas

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

  1. y=x2+4x−3
  2. y=2x2+10x+7
  3. y=−6x2+12x−1
  4. y=−x2+10x
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 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=x2+4x+7
  2. y=2x2−4x−2
  3. y=−3x2−18x−29
  4. y=−x2+12x−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 7 Graph Horizontal Parabolas

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

  1. x=2y2
  2. x=2y2+4y+6
  3. x=−y2+2y−4
  4. x=−3y2
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 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=4y2+8y
  2. x=y2+4y+5
  3. x=−y2−6y−7
  4. x=−2y2+4y
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 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=−19x2+103x

Ellipses

Exercise 10 Graph an Ellipse with Center at the Origin

In the following exercises, graph each ellipse.

  1. x236+y225=1
  2. x24+y281=1
  3. 49x2+64y2=3136
  4. 9x2+y2=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 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. x236+y264=1

Exercise 12 Graph an Ellipse with Center Not at the Origin

In the following exercises, graph each ellipse.

  1. (x−1)225+(y−6)24=1
  2. (x+4)216+(y+1)29=1
  3. (x−5)216+(y+3)236=1
  4. (x+3)29+(y−2)225=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 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. x2+y2+12x+40y+120=0
  2. 25x2+4y2−150x−56y+321=0
  3. 25x2+4y2+150x+125=0
  4. 4x2+9y2−126x+405=0
Answer

2.

  1. (x−3)24+(y−7)225=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. x29+(y−7)24=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 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 15 Graph a Hyperbola with Center at (0,0)

In the following exercises, graph.

  1. x225−y29=1
  2. y249−x216=1
  3. 9y2−16x2=144
  4. 16x2−4y2=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 16 Graph a Hyperbola with Center at (h,k)

In the following exercises, graph.

  1. (x+1)24−(y+1)29=1
  2. (x−2)24−(y−3)216=1
  3. (y+2)29−(x+1)29=1
  4. (y−1)225−(x−2)29=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 17 Graph a Hyperbola with Center at (h,k)

In the following exercises,

  1. Write the equation in standard form and
  2. Graph.
  1. 4x2−16y2+8x+96y−204=0
  2. 16x2−4y2−64x−24y−36=0
  3. 4y2−16x2+32x−8y−76=0
  4. 36y2−16x2−96x+216y−396=0
Answer

1.

  1. (x+1)216−(y−3)24=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. (y−1)216−(x−1)24=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 18 Identify the Graph of Each Equation as a Circle, Parabola, Ellipse, or Hyperbola

In the following exercises, identify the type of graph.

    1. 16y2−9x2−36x−96y−36=0
    2. x2+y2−4x+10y−7=0
    3. y=x2−2x+3
    4. 25x2+9y2=225
    1. x2+y2+4x−10y+25=0
    2. y2−x2−4y+2x−6=0
    3. x=−y2−2y+3
    4. 16x2+9y2=144
Answer

1.

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

Solve Systems of Nonlinear Equations

Exercise 19 Solve a System of Nonlinear Equations Using Graphing

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

  1. {3x2−y=0y=2x−1
  2. {y=x2−4y=x−4
  3. {x2+y2=169x=12
  4. {x2+y2=25y=−5
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 20 Solve a System of Nonlinear Equations Using Substitution

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

  1. {y=x2+3y=−2x+2
  2. {x2+y2=4x−y=4
  3. {9x2+4y2=36y−x=5
  4. {x2+4y2=42x−y=1
Answer

1. (−1,4)

3. No solution

Exercise 21 Solve a System of Nonlinear Equations Using Elimination

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

  1. {x2+y2=16x2−2y−1=0
  2. {x2−y2=5−2x2−3y2=−30
  3. {4x2+9y2=363y2−4x=12
  4. {x2+y2=14x2−y2=16
Answer

1. (−√7,3),(√7,3)

3. (−3,0),(0,−2),(0,2)

Exercise 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 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 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. x2+y2=121

3. (x+2)2+(y−3)2=52

Exercise 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. 4x2+49y2=196
  2. y=3(x−2)2−2
  3. 3x2+3y2=27
  4. y2100−x236=1
  5. x216+y281=1
  6. x=2y2+10y+7
  7. 64x2−9y2=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 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. 25x2+64y2+200x−256y−944=0
  2. x2+y2+10x+6y+30=0
  3. x=−y2+2y−4
  4. 9x2−25y2−36x−50y−214=0
  5. y=x2+6x+8
  6. Solve the nonlinear system of equations by graphing: {3y2−x=0y=−2x−1.
  7. Solve the nonlinear system of equations using substitution: {x2+y2=8y=−x−4.
  8. Solve the nonlinear system of equations using elimination: {x2+9y2=92x2−9y2=18
  9. Create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in y=ax2+bx+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. (x−2)225−(y+1)29=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. x22025+y21400=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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