22.2E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{23}\) Use the Distance Formula
In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.
- \((2,0)\) and \((5,4)\)
- \((-4,-3)\) and \((2,5)\)
- \((-4,-3)\) and \((8,2)\)
- \((-7,-3)\) and \((8,5)\)
- \((-1,4)\) and \((2,0)\)
- \((-1,3)\) and \((5,-5)\)
- \((1,-4)\) and \((6,8)\)
- \((-8,-2)\) and \((7,6)\)
- \((-3,-5)\) and \((0,1)\)
- \((-1,-2)\) and \((-3,4)\)
- \((3,-1)\) and \((1,7)\)
- \((-4,-5)\) and \((7,4)\)
- Answer
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1. \(d=5\)
3. \(13\)
5. \(5\)
7. \(13\)
9. \(d=3 \sqrt{5}, d \approx 6.7\)
11. \(d=\sqrt{68}, d \approx 8.2\)
Exercise \(\PageIndex{24}\) Use the Midpoint Formula
In the following exercises,
- find the midpoint of the line segments whose endpoints are given and
- plot the endpoints and the midpoint on a rectangular coordinate system.
- \((0,-5)\) and \((4,-3)\)
- \((-2,-6)\) and \((6,-2)\)
- \((3,-1)\) and \((4,-2)\)
- \((-3,-3)\) and \((6,-1)\)
- Answer
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1.
- Midpoint: \((2,-4)\)
3.
- Midpoint: \(\left(3 \frac{1}{2},-1 \frac{1}{2}\right)\)
Exercise \(\PageIndex{25}\) 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 radius and center \((0,0)\).
- Radius: \(7\)
- Radius: \(9\)
- Radius: \(\sqrt{2}\)
- Radius: \(\sqrt{5}\)
- Answer
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1. \(x^{2}+y^{2}=49\)
3. \(x^{2}+y^{2}=2\)
Exercise \(\PageIndex{26}\) 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 radius and center
- Radius: \(1\), center: \((3,5)\)
- Radius: \(10\), center: \((-2,6)\)
- Radius: \(2.5\), center: \((1.5, -3.5)\)
- Radius: \(1.5\), center: \((-5.5, -6.5)\)
- Answer
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1. \((x-3)^{2}+(y-5)^{2}=1\)
3. \((x-1.5)^{2}+(y+3.5)^{2}=6.25\)
Exercise \(\PageIndex{27}\) Write the Equation of a Circle in Standard Form
For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.
- Center \((3,−2)\) with point \((3,6)\)
- Center \((6,−6)\) with point \((2,−3)\)
- Center \((4,4)\) with point \((2,2)\)
- Center \((−5,6)\) with point \((−2,3)\)
- Answer
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1. \((x-3)^{2}+(y+2)^{2}=64\)
3. \((x-4)^{2}+(y-4)^{2}=8\)
Exercise \(\PageIndex{28}\) Graph a Circle
In the following exercises,
- find the center and radius, then
- graph each circle.
- \((x+5)^{2}+(y+3)^{2}=1\)
- \((x-2)^{2}+(y-3)^{2}=9\)
- \((x-4)^{2}+(y+2)^{2}=16\)
- \((x+2)^{2}+(y-5)^{2}=4\)
- \(x^{2}+(y+2)^{2}=25\)
- \((x-1)^{2}+y^{2}=36\)
- \((x-1.5)^{2}+(y+2.5)^{2}=0.25\)
- \((x-1)^{2}+(y-3)^{2}=\frac{9}{4}\)
- \(x^{2}+y^{2}=64\)
- \(x^{2}+y^{2}=49\)
- \(2 x^{2}+2 y^{2}=8\)
- \(6 x^{2}+6 y^{2}=216\)
- Answer
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1.
- The circle is centered at \((−5,−3)\) with a radius of \(1\).
3.
- The circle is centered at \((4,−2)\) with a radius of \(4\).
5.
- The circle is centered at \((0,−2)\) with a radius of \(5\).
7.
- The circle is centered at \((1.5,2.5)\) with a radius of \(0.5\).
9.
- The circle is centered at \((0,0)\) with a radius of \(8\).
11.
- The circle is centered at \((0,0)\) with a radius of \(2\).
Exercise \(\PageIndex{29}\) Graph a Circle
In the following exercises,
- identify the center and radius and
- graph.
- \(x^{2}+y^{2}+2 x+6 y+9=0\)
- \(x^{2}+y^{2}-6 x-8 y=0\)
- \(x^{2}+y^{2}-4 x+10 y-7=0\)
- \(x^{2}+y^{2}+12 x-14 y+21=0\)
- \(x^{2}+y^{2}+6 y+5=0\)
- \(x^{2}+y^{2}-10 y=0\)
- \(x^{2}+y^{2}+4 x=0\)
- \(x^{2}+y^{2}-14 x+13=0\)
- Answer
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1.
- Center: \((−1,−3)\), radius: \(1\)
3.
- Center: \((2,−5)\), radius: \(6\)
5.
- Center: \((0,−3)\), radius: \(2\)
7.
- Center: \((−2,0)\), radius: \(-2\)
Exercise \(\PageIndex{30}\) Writing Exercises
- Explain the relationship between the distance formula and the equation of a circle.
- Is a circle a function? Explain why or why not.
- In your own words, state the definition of a circle.
- In your own words, explain the steps you would take to change the general form of the equation of a circle to the standard form.
- Answer
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1. Answers will vary.
3. Answers will vary.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.