22.7: 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.
- \((-5,1)\) and \((-1,4)\)
- \((-2,5)\) and \((1,5)\)
- \((8,2)\) and \((-7,-3)\)
- \((1,-4)\) and \((5,-5)\)
- Answer
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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.
- \((-2,-6)\) and \((-4,-2)\)
- \((3,7)\) and \((5,1)\)
- \((-8,-10)\) and \((9,5)\)
- \((-3,2)\) and \((6,-9)\)
- Answer
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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.
- radius is \(15\) and center is \((0,0)\)
- radius is \(\sqrt{7}\) and center is \((0,0)\)
- radius is \(9\) and center is \((-3,5)\)
- radius is \(7\) and center is \((-2,-5)\)
- center is \((3,6)\) and a point on the circle is \((3,-2)\)
- center is \((2,2)\) and a point on the circle is \((4,4)\)
- Answer
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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,
- Find the center and radius, then
- Graph each circle.
- \(2 x^{2}+2 y^{2}=450\)
- \(3 x^{2}+3 y^{2}=432\)
- \((x+3)^{2}+(y-5)^{2}=81\)
- \((x+2)^{2}+(y+5)^{2}=49\)
- \(x^{2}+y^{2}-6 x-12 y-19=0\)
- \(x^{2}+y^{2}-4 y-60=0\)
- Answer
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2.
- radius: \(12,\) center: \((0,0)\)
4.
- radius: \(7,\) center: \((-2,-5)\)
6.
- radius: \(8,\) center: \((0,2)\)
Parabolas
Exercise \(\PageIndex{5}\) Graph Vertical Parabolas
In the following exercises, graph each equation by using its properties.
- \(y=x^{2}+4 x-3\)
- \(y=2 x^{2}+10 x+7\)
- \(y=-6 x^{2}+12 x-1\)
- \(y=-x^{2}+10 x\)
- Answer
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2.
4.
Exercise \(\PageIndex{6}\) Graph Vertical Parabolas
In the following exercises,
- Write the equation in standard form, then
- Use properties of the standard form to graph the equation.
- \(y=x^{2}+4 x+7\)
- \(y=2 x^{2}-4 x-2\)
- \(y=-3 x^{2}-18 x-29\)
- \(y=-x^{2}+12 x-35\)
- Answer
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2.
- \(y=2(x-1)^{2}-4\)
4.
- \(y=-(x-6)^{2}+1\)
Exercise \(\PageIndex{7}\) Graph Horizontal Parabolas
In the following exercises, graph each equation by using its properties.
- \(x=2 y^{2}\)
- \(x=2 y^{2}+4 y+6\)
- \(x=-y^{2}+2 y-4\)
- \(x=-3 y^{2}\)
- Answer
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2.
4.
Exercise \(\PageIndex{8}\) Graph Horizontal Parabolas
In the following exercises,
- Write the equation in standard form, then
- Use properties of the standard form to graph the equation.
- \(x=4 y^{2}+8 y\)
- \(x=y^{2}+4 y+5\)
- \(x=-y^{2}-6 y-7\)
- \(x=-2 y^{2}+4 y\)
- Answer
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2.
- \(x=(y+2)^{2}+1\)
4.
- \(x=-2(y-1)^{2}+2\)
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.
2.
- Answer
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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.
- \(\frac{x^{2}}{36}+\frac{y^{2}}{25}=1\)
- \(\frac{x^{2}}{4}+\frac{y^{2}}{81}=1\)
- \(49 x^{2}+64 y^{2}=3136\)
- \(9 x^{2}+y^{2}=9\)
- Answer
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2.
4.
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.
2.
- Answer
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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.
- \(\frac{(x-1)^{2}}{25}+\frac{(y-6)^{2}}{4}=1\)
- \(\frac{(x+4)^{2}}{16}+\frac{(y+1)^{2}}{9}=1\)
- \(\frac{(x-5)^{2}}{16}+\frac{(y+3)^{2}}{36}=1\)
- \(\frac{(x+3)^{2}}{9}+\frac{(y-2)^{2}}{25}=1\)
- Answer
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2.
4.
Exercise \(\PageIndex{13}\) Graph an Ellipse with Center Not at the Origin
In the following exercises,
- Write the equation in standard form and
- Graph.
- \(x^{2}+y^{2}+12 x+40 y+120=0\)
- \(25 x^{2}+4 y^{2}-150 x-56 y+321=0\)
- \(25 x^{2}+4 y^{2}+150 x+125=0\)
- \(4 x^{2}+9 y^{2}-126 x+405=0\)
- Answer
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2.
- \(\frac{(x-3)^{2}}{4}+\frac{(y-7)^{2}}{25}=1\)
4.
- \(\frac{x^{2}}{9}+\frac{(y-7)^{2}}{4}=1\)
Exercise \(\PageIndex{14}\) Solve Applications with Ellipses
In the following exercises, write the equation of the ellipse described.
- 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.
- Answer
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1. Solve
Hyperbolas
Exercise \(\PageIndex{15}\) Graph a Hyperbola with Center at \((0,0)\)
In the following exercises, graph.
- \(\frac{x^{2}}{25}-\frac{y^{2}}{9}=1\)
- \(\frac{y^{2}}{49}-\frac{x^{2}}{16}=1\)
- \(9 y^{2}-16 x^{2}=144\)
- \(16 x^{2}-4 y^{2}=64\)
- Answer
-
1.
3.
Exercise \(\PageIndex{16}\) Graph a Hyperbola with Center at \((h,k)\)
In the following exercises, graph.
- \(\frac{(x+1)^{2}}{4}-\frac{(y+1)^{2}}{9}=1\)
- \(\frac{(x-2)^{2}}{4}-\frac{(y-3)^{2}}{16}=1\)
- \(\frac{(y+2)^{2}}{9}-\frac{(x+1)^{2}}{9}=1\)
- \(\frac{(y-1)^{2}}{25}-\frac{(x-2)^{2}}{9}=1\)
- Answer
-
1.
3.
Exercise \(\PageIndex{17}\) Graph a Hyperbola with Center at \((h,k)\)
In the following exercises,
- Write the equation in standard form and
- Graph.
- \(4 x^{2}-16 y^{2}+8 x+96 y-204=0\)
- \(16 x^{2}-4 y^{2}-64 x-24 y-36=0\)
- \(4 y^{2}-16 x^{2}+32 x-8 y-76=0\)
- \(36 y^{2}-16 x^{2}-96 x+216 y-396=0\)
- Answer
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1.
- \(\frac{(x+1)^{2}}{16}-\frac{(y-3)^{2}}{4}=1\)
3.
- \(\frac{(y-1)^{2}}{16}-\frac{(x-1)^{2}}{4}=1\)
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.
-
- \(16 y^{2}-9 x^{2}-36 x-96 y-36=0\)
- \(x^{2}+y^{2}-4 x+10 y-7=0\)
- \(y=x^{2}-2 x+3\)
- \(25 x^{2}+9 y^{2}=225\)
-
- \(x^{2}+y^{2}+4 x-10 y+25=0\)
- \(y^{2}-x^{2}-4 y+2 x-6=0\)
- \(x=-y^{2}-2 y+3\)
- \(16 x^{2}+9 y^{2}=144\)
- Answer
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1.
- Hyperbola
- Circle
- Parabola
- 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.
- \(\left\{\begin{array}{l}{3 x^{2}-y=0} \\ {y=2 x-1}\end{array}\right.\)
- \(\left\{\begin{array}{l}{y=x^{2}-4} \\ {y=x-4}\end{array}\right.\)
- \(\left\{\begin{array}{l}{x^{2}+y^{2}=169} \\ {x=12}\end{array}\right.\)
- \(\left\{\begin{array}{l}{x^{2}+y^{2}=25} \\ {y=-5}\end{array}\right.\)
- Answer
-
1.
3.
Exercise \(\PageIndex{20}\) Solve a System of Nonlinear Equations Using Substitution
In the following exercises, solve the system of equations by using substitution.
- \(\left\{\begin{array}{l}{y=x^{2}+3} \\ {y=-2 x+2}\end{array}\right.\)
- \(\left\{\begin{array}{l}{x^{2}+y^{2}=4} \\ {x-y=4}\end{array}\right.\)
- \(\left\{\begin{array}{l}{9 x^{2}+4 y^{2}=36} \\ {y-x=5}\end{array}\right.\)
- \(\left\{\begin{array}{l}{x^{2}+4 y^{2}=4} \\ {2 x-y=1}\end{array}\right.\)
- Answer
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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.
- \(\left\{\begin{array}{l}{x^{2}+y^{2}=16} \\ {x^{2}-2 y-1=0}\end{array}\right.\)
- \(\left\{\begin{array}{l}{x^{2}-y^{2}=5} \\ {-2 x^{2}-3 y^{2}=-30}\end{array}\right.\)
- \(\left\{\begin{array}{l}{4 x^{2}+9 y^{2}=36} \\ {3 y^{2}-4 x=12}\end{array}\right.\)
- \(\left\{\begin{array}{l}{x^{2}+y^{2}=14} \\ {x^{2}-y^{2}=16}\end{array}\right.\)
- Answer
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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.
- The sum of the squares of two numbers is \(25\). The difference of the numbers is \(1\). Find the numbers.
- 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.
- The perimeter of a rectangle is \(58\) meters and its area is \(210\) square meters. Find the length and width of the rectangle.
- 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
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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.
- \((-4,-3)\) and \((-10,-11)\)
- \((6,8)\) and \((-5,-3)\)
- Answer
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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.
- radius is \(11\) and center is \((0,0)\)
- radius is \(12\) and center is \((10,-2)\)
- center is \((-2,3)\) and a point on the circle is \((2,-3)\)
- Find the equation of the ellipse shown in the graph.
- Answer
-
1. \(x^{2}+y^{2}=121\)
3. \((x+2)^{2}+(y-3)^{2}=52\)
Exercise \(\PageIndex{25}\)
In the following exercises,
- Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and
- Graph the equation.
- \(4 x^{2}+49 y^{2}=196\)
- \(y=3(x-2)^{2}-2\)
- \(3 x^{2}+3 y^{2}=27\)
- \(\frac{y^{2}}{100}-\frac{x^{2}}{36}=1\)
- \(\frac{x^{2}}{16}+\frac{y^{2}}{81}=1\)
- \(x=2 y^{2}+10 y+7\)
- \(64 x^{2}-9 y^{2}=576\)
- Answer
-
1.
- Ellipse
3.
- Circle
5.
- Ellipse
7.
- Hyperbola
Exercise \(\PageIndex{26}\)
In the following exercises,
- Identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola,
- Write the equation in standard form, and
- Graph the equation.
- \(25 x^{2}+64 y^{2}+200 x-256 y-944=0\)
- \(x^{2}+y^{2}+10 x+6 y+30=0\)
- \(x=-y^{2}+2 y-4\)
- \(9 x^{2}-25 y^{2}-36 x-50 y-214=0\)
- \(y=x^{2}+6 x+8\)
- Solve the nonlinear system of equations by graphing: \(\left\{\begin{array}{l}{3 y^{2}-x=0} \\ {y=-2 x-1}\end{array}\right.\).
- Solve the nonlinear system of equations using substitution: \(\left\{\begin{array}{l}{x^{2}+y^{2}=8} \\ {y=-x-4}\end{array}\right.\).
- Solve the nonlinear system of equations using elimination: \(\left\{\begin{array}{l}{x^{2}+9 y^{2}=9} \\ {2 x^{2}-9 y^{2}=18}\end{array}\right.\)
- 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.
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.
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.
- Circle
- \((x+5)^{2}+(y+3)^{2}=4\)
4.
- Hyperbola
- \(\frac{(x-2)^{2}}{25}-\frac{(y+1)^{2}}{9}=1\)
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.