{"id":16166,"date":"2020-03-02T00:21:21","date_gmt":"2020-03-02T00:21:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculus\/?post_type=chapter&#038;p=16166"},"modified":"2021-08-19T17:09:38","modified_gmt":"2021-08-19T17:09:38","slug":"circles","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/circles\/","title":{"raw":"Section 1.4: Circles","rendered":"Section 1.4: Circles"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Write the equation of a circle in standard form<\/li>\r\n \t<li>Graph a circle<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1165137533627\" class=\"note textbox\">\r\n<h3 class=\"title\">DEFINITION OF A CIRCLE<\/h3>\r\n<p id=\"fs-id1165135173375\">A <strong>circle<\/strong> is all points in a plane that are a fixed distance from a given point in the plane.\u00a0 The given point is called the <strong>center,<\/strong> [latex](h,k)[\/latex], and the fixed distance is called the <strong>radius<\/strong>, [latex]r[\/latex], of the circle.<\/p>\r\n\r\n<\/div>\r\n<div align=\"center\"><img src=\"http:\/\/www.hutchmath.com\/Images\/Circle.jpg\" alt=\"Picture of a Circle with center marked as (h,k) and radius r\" \/><\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>DERIVING THE STANDARD FORM OF A CIRCLE<\/h3>\r\n<div class=\"qa-wrapper\">\r\n\r\n<span style=\"font-size: 1rem;text-align: initial\">To derive the equation of a circle, we can use the distance formula with the points [latex](h,k)[\/latex], [latex](x,y)[\/latex], and the distance [latex]r[\/latex].<\/span>\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n<div style=\"text-align: center\">[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[\/latex]<\/div>\r\nSubstitute the values.\r\n<div style=\"text-align: center\">[latex]r=\\sqrt{(x-h)^{2}+(y-k)^{2}}[\/latex]<\/div>\r\nSquare both sides.\r\n<div style=\"text-align: center\">[latex]r^{2}=(x-h)^{2}+(y-k)^{2}[\/latex]<\/div>\r\n&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137533627\" class=\"note textbox\">\r\n<h3 class=\"title\">STANDARD FORM OF A CIRCLE<\/h3>\r\n<p id=\"fs-id1165137676320\">The <strong>standard form of a circle<\/strong> is as follows:<\/p>\r\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Write the Equation of a Circle in Standard Form<\/h2>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 1: WRITE THE STANDARD FORM Equation OF A CIRCLE<\/h3>\r\nWrite the standard form of a circle with radius [latex]3[\/latex] and center [latex](0,0)[\/latex].\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n\r\n&nbsp;\r\n\r\nUse the standard form of a circle.\r\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\nSubstitute in the values [latex]r=3, h=0, k=0[\/latex].\r\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-0\\right)}^{2}+{\\left(y-0\\right)}^{2}=3^{2}[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n\r\nSimplify.\r\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}=9[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>Example 2: WRITE THE STANDARD FORM equation OF A CIRCLE<\/h3>\r\nWrite the standard form of a circle with radius [latex]2[\/latex] and center [latex](-1,3)[\/latex].\r\n[reveal-answer q=\"380739\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380739\"]\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n\r\n&nbsp;\r\n\r\nUse the standard form of a circle.\r\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\nSubstitute in the values [latex]r=2, h=-1, k=3[\/latex].\r\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-(-1)\\right)}^{2}+{\\left(y-3\\right)}^{2}=2^{2}[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n\r\nSimplify.\r\n<p style=\"text-align: center\">[latex]{\\left(x+1\\right)}^{2}+{\\left(y-3\\right)}^{2}=4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div><\/div>\r\n<div><\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 3: finding the center and radius<\/h3>\r\nFind the center and radius, then graph the circle: [latex]{\\left(x+2\\right)}^{2}+{\\left(y-1\\right)}^{2}=9[\/latex].\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n\r\n&nbsp;\r\n\r\nUse the standard form of a circle.\r\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\nIdentify the center [latex](h,k)[\/latex], and radius [latex]r[\/latex].\r\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-(-2)\\right)}^{2}+{\\left(y-1\\right)}^{2}=3^{2}[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n\r\nThe center is\u00a0[latex](-2,1)[\/latex], and the radius is [latex]3[\/latex].\r\n\r\n&nbsp;\r\n\r\nNow graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 2 places.\r\n<div align=\"center\"><img src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx1.jpg\" alt=\"Graph of a Circle with center (-2,1) and radius 3\" \/><\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 4: finding the center and radius<\/h3>\r\nFind the center and radius, then graph the circle: [latex]{4x}^{2}+{4y}^{2}=64[\/latex].\r\n[reveal-answer q=\"380740\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380740\"]\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n<div style=\"text-align: center\"><\/div>\r\n<\/div>\r\n&nbsp;\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n\r\nDivide each side by 4.\r\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/p>\r\nUse the standard form of a circle.\r\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\nIdentify the center [latex](h,k)[\/latex], and radius [latex]r[\/latex].\r\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-0\\right)}^{2}+{\\left(y-0\\right)}^{2}=4^{2}[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n\r\nThe center is\u00a0[latex](0,0)[\/latex], and the radius is [latex]4[\/latex].\r\n\r\n&nbsp;\r\n\r\nNow graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 4 places.\r\n<div align=\"center\"><img src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx2.jpg\" alt=\"Graph of a Circle with center (0,0) and radius 4\" \/><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1165137533627\" class=\"note textbox\">\r\n<h3 class=\"title\">GENERAL FORM OF A CIRCLE<\/h3>\r\n<p id=\"fs-id1165137676320\">The <strong>general form of a circle<\/strong> is as follows:<\/p>\r\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}+ax+by+c=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 5: WRITE THE STANDARD FORM Equation OF A CIRCLE<\/h3>\r\nFind the center and radius, then graph: [latex]{x}^{2}+{y}^{2}-4x-6y+4=0[\/latex].\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n<div style=\"text-align: center\"><\/div>\r\nWe need to rewrite this general form into standard form in order to find the center and radius.\r\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}-4x-6y+4=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\nGroup the x-terms and y-terms.\u00a0 Collect the constants on the right right side.\r\n<p style=\"text-align: center\">\u00a0 [latex]{x}^{2}-4x+{y}^{2}-6y=-4[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n\r\nComplete the squares.\r\n<p style=\"text-align: center\">[latex]{x}^{2}-4x+4+{y}^{2}-6y+9=-4+4+9[\/latex]<\/p>\r\n\r\n<\/div>\r\nRewrite as binomial squares.\r\n<p style=\"text-align: center\">[latex]{\\left(x-2\\right)}^{2}+{\\left(y-3\\right)}^{2}=9[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe center is\u00a0[latex](2,3)[\/latex], and the radius is [latex]3[\/latex].\r\n\r\n&nbsp;\r\n\r\nNow graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 3 places.\r\n<div align=\"center\"><img src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx3.jpg\" alt=\"Graph of a Circle with center centered at (2,3) and radius 3\" \/><\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 6: WRITE THE STANDARD FORM Equation OF A CIRCLE<\/h3>\r\nFind the center and radius, then graph: [latex]{x}^{2}+{y}^{2}+8y=0[\/latex].\r\n[reveal-answer q=\"380741\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"380741\"]\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n<div style=\"text-align: center\"><\/div>\r\nWe need to rewrite this general form into standard form in order to find the center and radius.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}+8y=0[\/latex]<\/p>\r\n\r\n<\/div>\r\nGroup the x-terms and y-terms.\r\n<p style=\"text-align: center\">\u00a0 [latex]{x}^{2}+{y}^{2}+8y=0[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n\r\nComplete the squares.\r\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}+8y+16=0+16[\/latex]<\/p>\r\n\r\n<\/div>\r\nRewrite as binomial squares.\r\n<p style=\"text-align: center\">[latex]{\\left(x-0\\right)}^{2}+{\\left(y+4\\right)}^{2}=16[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe center is\u00a0[latex](0,-4)[\/latex], and the radius is [latex]4[\/latex].\r\n\r\n&nbsp;\r\n\r\nNow graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 4 places.\r\n<div align=\"center\"><img src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx4.jpg\" alt=\"Graph of a Circle with center centered at (0,-4) and radius 4\" \/><\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 7: APPLYING THE DISTANCE AND MIDPOINT FORMULAS TO A CIRCLE EQUATION<\/h3>\r\nThe diameter of a circle has endpoints [latex](-1,-4)[\/latex] and [latex](7,2)[\/latex]. Find the center and radius of the circle and also write its standard form equation.\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n<div style=\"text-align: center\"><\/div>\r\nThe center of a circle is the center, or midpoint, of its diameter.\u00a0 Thus the midpoint formula will yield the center point.\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<p style=\"text-align: center\">\u00a0 [latex]=\\left(\\frac{-1+7}{2},\\frac{-4+2}{2}\\right)[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n<p style=\"text-align: center\">[latex]=\\left(\\frac{6}{2},\\frac{-2}{2}\\right)[\/latex]<\/p>\r\n\r\n<\/div>\r\n<p style=\"text-align: center\">[latex]=(3,-1)[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe center is\u00a0[latex](3,-1)[\/latex].\u00a0 The distance formula will be used to find the distance from the center to one of the points on the circle.\u00a0 This will yield the radius:\r\n\r\n&nbsp;\r\n<p style=\"text-align: center\">[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]d=\\sqrt{(7-3)^{2}+(2-(-1))^{2}}[\/latex]<\/p>\r\n<p style=\"text-align: center\">[latex]d=\\sqrt{4^{2}+3^{2}}=5[\/latex]<\/p>\r\n&nbsp;\r\n\r\nThe distance from the center to a point on the circle is 5.\u00a0 Therefore the radius is 5.\u00a0 The center and radius can now be used to find the standard form of the circle:\r\n\r\n&nbsp;\r\n<div id=\"q676921\" class=\"hidden-answer\">\r\n\r\nStart with the standard form of a circle.\r\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\r\n\r\n<\/div>\r\nSubstitute in the values [latex]r=5, h=3, k=-1[\/latex].\r\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-3\\right)}^{2}+{\\left(y-(-1)\\right)}^{2}=5^{2}[\/latex]<\/p>\r\n\r\n<div id=\"q676921\">\r\n\r\nSimplify.\r\n<p style=\"text-align: center\">[latex]{\\left(x-3\\right)}^{2}+{\\left(y+1\\right)}^{2}=25[\/latex]<\/p>\r\n\r\n<\/div>\r\n&nbsp;\r\n\r\n&nbsp;\r\n<div align=\"center\"><\/div>\r\n<\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example 8: Finding the Center of a Circle<\/h3>\r\nThe diameter of a circle has endpoints [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(5,-4\\right)[\/latex]. Find the center of the circle.\r\n\r\n[reveal-answer q=\"418175\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418175\"]\r\n\r\nThe center of a circle is the center or midpoint of its diameter. Thus, the midpoint formula will yield the center point.\r\n<div style=\"text-align: center\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4 - 4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Key Concepts<\/span><\/h2>\r\n<ul id=\"fs-id1165137851183\">\r\n \t<li>A circle is all points in a plane that are a fixed distance from a given point on the plane.\u00a0 The given point is called the center, and the fixed distance is called the radius.<\/li>\r\n \t<li>The standard form of the equation of a circle with center [latex](h,k)[\/latex] and radius [latex]r[\/latex] is\u00a0[latex]{\\left(x-h\\right)}^{2}{+}{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<dl id=\"fs-id1165135315542\" class=\"definition\">\r\n \t<dd><\/dd>\r\n<\/dl>\r\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Section 1.2 Homework Exercises<\/span><\/h2>\r\n<\/div>\r\nFor the following exercises, write the standard form of the equation of the circle with the given radius and center [latex](0,0)[\/latex].\r\n\r\n1. Radius: [latex]7[\/latex]\r\n\r\n2. Radius: [latex]9[\/latex]\r\n\r\n3. Radius: [latex]\\sqrt{2}[\/latex]\r\n\r\n4. Radius: [latex]\\sqrt{5}[\/latex]\r\n\r\nIn the following exercises, write the standard form of the equation of the circle with the given radius and center.\r\n\r\n5. Radius: [latex]1[\/latex], center: [latex](3,5)[\/latex]\r\n\r\n6. Radius: [latex]10[\/latex], center: [latex](-2,6)[\/latex]\r\n\r\n7. Radius: [latex]2.5[\/latex], center: [latex](1.5,-3.5)[\/latex]\r\n\r\n8. Radius: [latex]1.5[\/latex], center: [latex](-5.5,-6.5)[\/latex]\r\n\r\nFor the following exercises, write the standard form of the equation of the circle with the given center and point on the circle.\r\n\r\n9. Center: [latex](3,-2)[\/latex] with point [latex](3,6)[\/latex]\r\n\r\n10. Center: [latex](6,-6)[\/latex] with point [latex](2,-3)[\/latex]\r\n\r\n11. Center: [latex](4,4)[\/latex] with point [latex](2,2)[\/latex]\r\n\r\n12. Center: [latex](-5,6)[\/latex] with point [latex](-2,3)[\/latex]\r\n\r\nIn the following exercises, find the center and radius and then graph each circle.\r\n\r\n13. [latex]{\\left(x+5\\right)}^{2}+{\\left(y+3\\right)}^{2}=1[\/latex]\r\n\r\n14. [latex]{\\left(x-2\\right)}^{2}+{\\left(y-3\\right)}^{2}=9[\/latex]\r\n\r\n15. [latex]{\\left(x-4\\right)}^{2}+{\\left(y+2\\right)}^{2}=16[\/latex]\r\n\r\n16. [latex]{\\left(x+2\\right)}^{2}+{\\left(y-5\\right)}^{2}=4[\/latex]\r\n\r\n17. [latex]x^{2}+{\\left(y+2\\right)}^{2}=25[\/latex]\r\n\r\n18. [latex]{\\left(x-1\\right)}^{2}+{y}^{2}=36[\/latex]\r\n\r\n19. [latex]{\\left(x-1.5\\right)}^{2}+{\\left(y-2.5\\right)}^{2}=0.25[\/latex]\r\n\r\n20. [latex]{\\left(x-1\\right)}^{2}+{\\left(y-3\\right)}^{2}=\\frac{9}{4}[\/latex]\r\n\r\n21. [latex]x^{2}+y^{2}=64[\/latex]\r\n\r\n22. [latex]x^{2}+y^{2}=49[\/latex]\r\n\r\n23. [latex]2x^{2}+2y^{2}=8[\/latex]\r\n\r\n24. [latex]6x^{2}+6y^{2}=216[\/latex]\r\n\r\nIn the following exercises, identify the center and radius and graph.\r\n\r\n25. [latex]x^{2}+y^{2}+2x+6y+9=0[\/latex]\r\n\r\n26. [latex]x^{2}+y^{2}-6x-8y=0[\/latex]\r\n\r\n27. [latex]x^{2}+y^{2}-4x+10y-7=0[\/latex]\r\n\r\n28. [latex]x^{2}+y^{2}+12x-14y+21=0[\/latex]\r\n\r\n29. [latex]x^{2}+y^{2}+6y+5=0[\/latex]\r\n\r\n30. [latex]x^{2}+y^{2}-10y=0[\/latex]\r\n\r\n31. [latex]x^{2}+y^{2}+4x=0[\/latex]\r\n\r\n32. [latex]x^{2}+y^{2}-14x+13=0[\/latex]\r\n\r\n33. Explain the relationship between the distance formula and the equation of a circle.\r\n\r\n34. In your own words, state the definition of a circle.\r\n\r\n35. 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.","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Write the equation of a circle in standard form<\/li>\n<li>Graph a circle<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165137533627\" class=\"note textbox\">\n<h3 class=\"title\">DEFINITION OF A CIRCLE<\/h3>\n<p id=\"fs-id1165135173375\">A <strong>circle<\/strong> is all points in a plane that are a fixed distance from a given point in the plane.\u00a0 The given point is called the <strong>center,<\/strong> [latex](h,k)[\/latex], and the fixed distance is called the <strong>radius<\/strong>, [latex]r[\/latex], of the circle.<\/p>\n<\/div>\n<div style=\"margin: auto;\"><img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/Circle.jpg\" alt=\"Picture of a Circle with center marked as (h,k) and radius r\" \/><\/div>\n<div class=\"textbox exercises\">\n<h3>DERIVING THE STANDARD FORM OF A CIRCLE<\/h3>\n<div class=\"qa-wrapper\">\n<p><span style=\"font-size: 1rem;text-align: initial\">To derive the equation of a circle, we can use the distance formula with the points [latex](h,k)[\/latex], [latex](x,y)[\/latex], and the distance [latex]r[\/latex].<\/span><\/p>\n<div id=\"q676921\" class=\"hidden-answer\">\n<div style=\"text-align: center\">[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[\/latex]<\/div>\n<p>Substitute the values.<\/p>\n<div style=\"text-align: center\">[latex]r=\\sqrt{(x-h)^{2}+(y-k)^{2}}[\/latex]<\/div>\n<p>Square both sides.<\/p>\n<div style=\"text-align: center\">[latex]r^{2}=(x-h)^{2}+(y-k)^{2}[\/latex]<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137533627\" class=\"note textbox\">\n<h3 class=\"title\">STANDARD FORM OF A CIRCLE<\/h3>\n<p id=\"fs-id1165137676320\">The <strong>standard form of a circle<\/strong> is as follows:<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\n<\/div>\n<h2><\/h2>\n<h2>Write the Equation of a Circle in Standard Form<\/h2>\n<div class=\"textbox exercises\">\n<h3>Example 1: WRITE THE STANDARD FORM Equation OF A CIRCLE<\/h3>\n<p>Write the standard form of a circle with radius [latex]3[\/latex] and center [latex](0,0)[\/latex].<\/p>\n<div id=\"q676921\" class=\"hidden-answer\">\n<p>&nbsp;<\/p>\n<p>Use the standard form of a circle.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\n<\/div>\n<p>Substitute in the values [latex]r=3, h=0, k=0[\/latex].<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-0\\right)}^{2}+{\\left(y-0\\right)}^{2}=3^{2}[\/latex]<\/p>\n<div id=\"q676921\">\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}=9[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>Example 2: WRITE THE STANDARD FORM equation OF A CIRCLE<\/h3>\n<p>Write the standard form of a circle with radius [latex]2[\/latex] and center [latex](-1,3)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380739\">Show Solution<\/span><\/p>\n<div id=\"q380739\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"q676921\" class=\"hidden-answer\">\n<p>&nbsp;<\/p>\n<p>Use the standard form of a circle.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\n<\/div>\n<p>Substitute in the values [latex]r=2, h=-1, k=3[\/latex].<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-(-1)\\right)}^{2}+{\\left(y-3\\right)}^{2}=2^{2}[\/latex]<\/p>\n<div id=\"q676921\">\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x+1\\right)}^{2}+{\\left(y-3\\right)}^{2}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<div><\/div>\n<div class=\"textbox exercises\">\n<h3>Example 3: finding the center and radius<\/h3>\n<p>Find the center and radius, then graph the circle: [latex]{\\left(x+2\\right)}^{2}+{\\left(y-1\\right)}^{2}=9[\/latex].<\/p>\n<div id=\"q676921\" class=\"hidden-answer\">\n<p>&nbsp;<\/p>\n<p>Use the standard form of a circle.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\n<\/div>\n<p>Identify the center [latex](h,k)[\/latex], and radius [latex]r[\/latex].<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-(-2)\\right)}^{2}+{\\left(y-1\\right)}^{2}=3^{2}[\/latex]<\/p>\n<div id=\"q676921\">\n<p>The center is\u00a0[latex](-2,1)[\/latex], and the radius is [latex]3[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Now graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 2 places.<\/p>\n<div style=\"margin: auto;\"><img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx1.jpg\" alt=\"Graph of a Circle with center (-2,1) and radius 3\" \/><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 4: finding the center and radius<\/h3>\n<p>Find the center and radius, then graph the circle: [latex]{4x}^{2}+{4y}^{2}=64[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380740\">Show Solution<\/span><\/p>\n<div id=\"q380740\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"q676921\" class=\"hidden-answer\">\n<div style=\"text-align: center\"><\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div id=\"q676921\" class=\"hidden-answer\">\n<p>Divide each side by 4.<\/p>\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/p>\n<p>Use the standard form of a circle.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\n<\/div>\n<p>Identify the center [latex](h,k)[\/latex], and radius [latex]r[\/latex].<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-0\\right)}^{2}+{\\left(y-0\\right)}^{2}=4^{2}[\/latex]<\/p>\n<div id=\"q676921\">\n<p>The center is\u00a0[latex](0,0)[\/latex], and the radius is [latex]4[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Now graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 4 places.<\/p>\n<div style=\"margin: auto;\"><img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx2.jpg\" alt=\"Graph of a Circle with center (0,0) and radius 4\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137533627\" class=\"note textbox\">\n<h3 class=\"title\">GENERAL FORM OF A CIRCLE<\/h3>\n<p id=\"fs-id1165137676320\">The <strong>general form of a circle<\/strong> is as follows:<\/p>\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}+ax+by+c=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 5: WRITE THE STANDARD FORM Equation OF A CIRCLE<\/h3>\n<p>Find the center and radius, then graph: [latex]{x}^{2}+{y}^{2}-4x-6y+4=0[\/latex].<\/p>\n<div id=\"q676921\" class=\"hidden-answer\">\n<div style=\"text-align: center\"><\/div>\n<p>We need to rewrite this general form into standard form in order to find the center and radius.<\/p>\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}-4x-6y+4=0[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Group the x-terms and y-terms.\u00a0 Collect the constants on the right right side.<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]{x}^{2}-4x+{y}^{2}-6y=-4[\/latex]<\/p>\n<div id=\"q676921\">\n<p>Complete the squares.<\/p>\n<p style=\"text-align: center\">[latex]{x}^{2}-4x+4+{y}^{2}-6y+9=-4+4+9[\/latex]<\/p>\n<\/div>\n<p>Rewrite as binomial squares.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-2\\right)}^{2}+{\\left(y-3\\right)}^{2}=9[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The center is\u00a0[latex](2,3)[\/latex], and the radius is [latex]3[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Now graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 3 places.<\/p>\n<div style=\"margin: auto;\"><img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx3.jpg\" alt=\"Graph of a Circle with center centered at (2,3) and radius 3\" \/><\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 6: WRITE THE STANDARD FORM Equation OF A CIRCLE<\/h3>\n<p>Find the center and radius, then graph: [latex]{x}^{2}+{y}^{2}+8y=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q380741\">Show Solution<\/span><\/p>\n<div id=\"q380741\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"q676921\" class=\"hidden-answer\">\n<div style=\"text-align: center\"><\/div>\n<p>We need to rewrite this general form into standard form in order to find the center and radius.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}+8y=0[\/latex]<\/p>\n<\/div>\n<p>Group the x-terms and y-terms.<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]{x}^{2}+{y}^{2}+8y=0[\/latex]<\/p>\n<div id=\"q676921\">\n<p>Complete the squares.<\/p>\n<p style=\"text-align: center\">[latex]{x}^{2}+{y}^{2}+8y+16=0+16[\/latex]<\/p>\n<\/div>\n<p>Rewrite as binomial squares.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-0\\right)}^{2}+{\\left(y+4\\right)}^{2}=16[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The center is\u00a0[latex](0,-4)[\/latex], and the radius is [latex]4[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p>Now graph the circle.\u00a0 Plot the center first and then go up, down, left, and right 4 places.<\/p>\n<div style=\"margin: auto;\"><img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/CircleGraphEx4.jpg\" alt=\"Graph of a Circle with center centered at (0,-4) and radius 4\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example 7: APPLYING THE DISTANCE AND MIDPOINT FORMULAS TO A CIRCLE EQUATION<\/h3>\n<p>The diameter of a circle has endpoints [latex](-1,-4)[\/latex] and [latex](7,2)[\/latex]. Find the center and radius of the circle and also write its standard form equation.<\/p>\n<div id=\"q676921\" class=\"hidden-answer\">\n<div style=\"text-align: center\"><\/div>\n<p>The center of a circle is the center, or midpoint, of its diameter.\u00a0 Thus the midpoint formula will yield the center point.<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[latex]M=\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)[\/latex]<\/p>\n<\/div>\n<p style=\"text-align: center\">\u00a0 [latex]=\\left(\\frac{-1+7}{2},\\frac{-4+2}{2}\\right)[\/latex]<\/p>\n<div id=\"q676921\">\n<p style=\"text-align: center\">[latex]=\\left(\\frac{6}{2},\\frac{-2}{2}\\right)[\/latex]<\/p>\n<\/div>\n<p style=\"text-align: center\">[latex]=(3,-1)[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The center is\u00a0[latex](3,-1)[\/latex].\u00a0 The distance formula will be used to find the distance from the center to one of the points on the circle.\u00a0 This will yield the radius:<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center\">[latex]d=\\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]d=\\sqrt{(7-3)^{2}+(2-(-1))^{2}}[\/latex]<\/p>\n<p style=\"text-align: center\">[latex]d=\\sqrt{4^{2}+3^{2}}=5[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The distance from the center to a point on the circle is 5.\u00a0 Therefore the radius is 5.\u00a0 The center and radius can now be used to find the standard form of the circle:<\/p>\n<p>&nbsp;<\/p>\n<div id=\"q676921\" class=\"hidden-answer\">\n<p>Start with the standard form of a circle.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-h\\right)}^{2}+{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/p>\n<\/div>\n<p>Substitute in the values [latex]r=5, h=3, k=-1[\/latex].<\/p>\n<p style=\"text-align: center\">\u00a0 [latex]{\\left(x-3\\right)}^{2}+{\\left(y-(-1)\\right)}^{2}=5^{2}[\/latex]<\/p>\n<div id=\"q676921\">\n<p>Simplify.<\/p>\n<p style=\"text-align: center\">[latex]{\\left(x-3\\right)}^{2}+{\\left(y+1\\right)}^{2}=25[\/latex]<\/p>\n<\/div>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<div style=\"margin: auto;\"><\/div>\n<\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example 8: Finding the Center of a Circle<\/h3>\n<p>The diameter of a circle has endpoints [latex]\\left(-1,-4\\right)[\/latex] and [latex]\\left(5,-4\\right)[\/latex]. Find the center of the circle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q418175\">Show Solution<\/span><\/p>\n<div id=\"q418175\" class=\"hidden-answer\" style=\"display: none\">\n<p>The center of a circle is the center or midpoint of its diameter. Thus, the midpoint formula will yield the center point.<\/p>\n<div style=\"text-align: center\">[latex]\\begin{array}{l}\\left(\\frac{{x}_{1}+{x}_{2}}{2},\\frac{{y}_{1}+{y}_{2}}{2}\\right)\\\\ \\left(\\frac{-1+5}{2},\\frac{-4 - 4}{2}\\right)=\\left(\\frac{4}{2},-\\frac{8}{2}\\right)=\\left(2,-4\\right)\\end{array}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Key Concepts<\/span><\/h2>\n<ul id=\"fs-id1165137851183\">\n<li>A circle is all points in a plane that are a fixed distance from a given point on the plane.\u00a0 The given point is called the center, and the fixed distance is called the radius.<\/li>\n<li>The standard form of the equation of a circle with center [latex](h,k)[\/latex] and radius [latex]r[\/latex] is\u00a0[latex]{\\left(x-h\\right)}^{2}{+}{\\left(y-k\\right)}^{2}=r^{2}[\/latex]<\/li>\n<\/ul>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n<dd><\/dd>\n<\/dl>\n<h2 style=\"text-align: center\"><span style=\"text-decoration: underline\">Section 1.2 Homework Exercises<\/span><\/h2>\n<\/div>\n<p>For the following exercises, write the standard form of the equation of the circle with the given radius and center [latex](0,0)[\/latex].<\/p>\n<p>1. Radius: [latex]7[\/latex]<\/p>\n<p>2. Radius: [latex]9[\/latex]<\/p>\n<p>3. Radius: [latex]\\sqrt{2}[\/latex]<\/p>\n<p>4. Radius: [latex]\\sqrt{5}[\/latex]<\/p>\n<p>In the following exercises, write the standard form of the equation of the circle with the given radius and center.<\/p>\n<p>5. Radius: [latex]1[\/latex], center: [latex](3,5)[\/latex]<\/p>\n<p>6. Radius: [latex]10[\/latex], center: [latex](-2,6)[\/latex]<\/p>\n<p>7. Radius: [latex]2.5[\/latex], center: [latex](1.5,-3.5)[\/latex]<\/p>\n<p>8. Radius: [latex]1.5[\/latex], center: [latex](-5.5,-6.5)[\/latex]<\/p>\n<p>For the following exercises, write the standard form of the equation of the circle with the given center and point on the circle.<\/p>\n<p>9. Center: [latex](3,-2)[\/latex] with point [latex](3,6)[\/latex]<\/p>\n<p>10. Center: [latex](6,-6)[\/latex] with point [latex](2,-3)[\/latex]<\/p>\n<p>11. Center: [latex](4,4)[\/latex] with point [latex](2,2)[\/latex]<\/p>\n<p>12. Center: [latex](-5,6)[\/latex] with point [latex](-2,3)[\/latex]<\/p>\n<p>In the following exercises, find the center and radius and then graph each circle.<\/p>\n<p>13. [latex]{\\left(x+5\\right)}^{2}+{\\left(y+3\\right)}^{2}=1[\/latex]<\/p>\n<p>14. [latex]{\\left(x-2\\right)}^{2}+{\\left(y-3\\right)}^{2}=9[\/latex]<\/p>\n<p>15. [latex]{\\left(x-4\\right)}^{2}+{\\left(y+2\\right)}^{2}=16[\/latex]<\/p>\n<p>16. [latex]{\\left(x+2\\right)}^{2}+{\\left(y-5\\right)}^{2}=4[\/latex]<\/p>\n<p>17. [latex]x^{2}+{\\left(y+2\\right)}^{2}=25[\/latex]<\/p>\n<p>18. [latex]{\\left(x-1\\right)}^{2}+{y}^{2}=36[\/latex]<\/p>\n<p>19. [latex]{\\left(x-1.5\\right)}^{2}+{\\left(y-2.5\\right)}^{2}=0.25[\/latex]<\/p>\n<p>20. [latex]{\\left(x-1\\right)}^{2}+{\\left(y-3\\right)}^{2}=\\frac{9}{4}[\/latex]<\/p>\n<p>21. [latex]x^{2}+y^{2}=64[\/latex]<\/p>\n<p>22. [latex]x^{2}+y^{2}=49[\/latex]<\/p>\n<p>23. [latex]2x^{2}+2y^{2}=8[\/latex]<\/p>\n<p>24. [latex]6x^{2}+6y^{2}=216[\/latex]<\/p>\n<p>In the following exercises, identify the center and radius and graph.<\/p>\n<p>25. [latex]x^{2}+y^{2}+2x+6y+9=0[\/latex]<\/p>\n<p>26. [latex]x^{2}+y^{2}-6x-8y=0[\/latex]<\/p>\n<p>27. [latex]x^{2}+y^{2}-4x+10y-7=0[\/latex]<\/p>\n<p>28. [latex]x^{2}+y^{2}+12x-14y+21=0[\/latex]<\/p>\n<p>29. [latex]x^{2}+y^{2}+6y+5=0[\/latex]<\/p>\n<p>30. [latex]x^{2}+y^{2}-10y=0[\/latex]<\/p>\n<p>31. [latex]x^{2}+y^{2}+4x=0[\/latex]<\/p>\n<p>32. [latex]x^{2}+y^{2}-14x+13=0[\/latex]<\/p>\n<p>33. Explain the relationship between the distance formula and the equation of a circle.<\/p>\n<p>34. In your own words, state the definition of a circle.<\/p>\n<p>35. 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.<\/p>\n","protected":false},"author":264444,"menu_order":4,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-16166","chapter","type-chapter","status-publish","hentry"],"part":10705,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16166","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":94,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16166\/revisions"}],"predecessor-version":[{"id":17683,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16166\/revisions\/17683"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/10705"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/16166\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=16166"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=16166"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=16166"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=16166"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}