{"id":2555,"date":"2024-08-05T23:21:39","date_gmt":"2024-08-05T23:21:39","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=2555"},"modified":"2025-08-15T16:38:06","modified_gmt":"2025-08-15T16:38:06","slug":"circles-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/circles-fresh-take\/","title":{"raw":"Circles: Fresh Take","rendered":"Circles: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Write the equations for circles using the standard form<\/li>\r\n \t<li>Graph a circle<\/li>\r\n \t<li>Solve system of equations involving circles<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Write the Equation of a Circle in Standard Form<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Circle Definition:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Set of points equidistant from a center point in a plane<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](h,k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Radius: [latex]r[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Standard Form Equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](x-h)^2+(y-k)^2 = r^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex](h,k)[\/latex] represents the center<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is the radius<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Components of the Equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](x-h)^2[\/latex]: horizontal distance from center<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex](y-k)^2[\/latex]: vertical distance from center<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Right side [latex]r^2[\/latex]: squared radius<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Variations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center at origin: [latex]x^2+y^2 = r^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Negative values inside parentheses change to addition<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Write the equation of a circle with center [latex](-2, 5)[\/latex] and radius [latex]4[\/latex].[reveal-answer q=\"822019\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"822019\"]Identify the values:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex]h = -2[\/latex], [latex]k = 5[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Radius: [latex]r = 4[\/latex]<\/li>\r\n<\/ul>\r\nPlug these values into the standard form:\r\n\r\n<center>[latex](x-(-2))^2+(y-(5))^2 = 4^2[\/latex]<\/center>\r\nSimplify:\r\n\r\n<center>[latex](x+2)^2+(y-5)^2 = 16[\/latex]<\/center>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aedhdgga-Xmde2JLKAFQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Xmde2JLKAFQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-aedhdgga-Xmde2JLKAFQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851177&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aedhdgga-Xmde2JLKAFQ&vembed=0&video_id=Xmde2JLKAFQ&video_target=tpm-plugin-aedhdgga-Xmde2JLKAFQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Write+General+Equation+of+a+Circle+in+Standard+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Write General Equation of a Circle in Standard Form\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2 data-type=\"title\">Graph a Circle<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Equation to Graph Relationship:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Standard form: [latex](x-h)^2 + (y-k)^2 = r^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Each part of the equation corresponds to a graphical feature<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Components for Graphing:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](h,k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Radius: [latex]r[\/latex] (square root of the right side)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Critical Points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](h,k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Top: [latex](h, k+r)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Bottom: [latex](h, k-r)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Right: [latex](h+r, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Left: [latex](h-r, k)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Symmetry:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Circles are symmetrical about their center<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical and horizontal lines through the center are lines of symmetry<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the center and radius, then graph the circle:<center>[latex](x+2)^2+(y\u22121)^2=9[\/latex]<\/center>[reveal-answer q=\"738682\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"738682\"]Use the standard form of the equation of a circle. Identify the center, [latex](h,k)[\/latex] and radius, [latex]r[\/latex].\r\n<p style=\"text-align: center;\">[latex](x+2)^2+(y\u22121)^2=9[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex](x-(-2))^2 + (y-(1))^2 = 3^2[\/latex]<\/p>\r\n<strong>Thus, the center is [latex](-2,1)[\/latex] and the radius is [latex]3[\/latex].<\/strong>\r\n\r\nGraph the circle:\r\n\r\n[caption id=\"attachment_5238\" align=\"aligncenter\" width=\"318\"]<img class=\"wp-image-5238 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14134957\/22466db6b6d5f98c6597ff0004e6085ee3b4255b.jpg\" alt=\"a coordinate plane with a circle centered at the point (-2, 1). The circle has a radius of 3, indicated by a labeled line segment from the center to a point on the circle's circumference.\" width=\"318\" height=\"323\" \/> Circle on a coordinate plane with the center and radius labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-pre-wrap break-words\">Graph the circle given by the equation: [latex](x+1)^2 + (y-3)^2 = 9[\/latex]<\/p>\r\n[reveal-answer q=\"981055\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"981055\"]\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Identify center and radius:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Rewrite in standard form: [latex](x-(-1))^2 + (y-3)^2 = 3^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](-1,3)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Radius: [latex]r = 3[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plot the center at [latex](-1, 3)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Mark the four key points:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Top: [latex](-1, 6)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Bottom: [latex](-1, 0)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Right: [latex](2, 3)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Left: [latex](-4, 3)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Connect these points with a smooth circular curve\r\n\r\n[caption id=\"attachment_5239\" align=\"aligncenter\" width=\"681\"]<img class=\"wp-image-5239 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14140126\/Screenshot-2024-10-14-100106.png\" alt=\"a coordinate plane with a red circle centered at the point (-1, 3). The radius of the circle is indicated by several labeled points on its circumference: (-1, 6), (-1, 0), (2, 3), and (-4, 3).\" width=\"681\" height=\"637\" \/> Circle on a coordinate plane with the center and four points labeled[\/caption]\r\n\r\n[\/hidden-answer]<\/li>\r\n<\/ul>\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddhbcdeb-dm85p_X_L4A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/dm85p_X_L4A?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-ddhbcdeb-dm85p_X_L4A\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851178&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ddhbcdeb-dm85p_X_L4A&vembed=0&video_id=dm85p_X_L4A&video_target=tpm-plugin-ddhbcdeb-dm85p_X_L4A'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Circle+-+Write+the+Equation+in+Standard+form+x%5E2%2By%5E2-10y%2B16%3D0_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Circle: Write the Equation in Standard form x^2+y^2-10y+16=0\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>General Form of the Equation of a Circle<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">General Form Equation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex]x^2 + y^2 + ax + by + c = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are constants<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Does not directly show center or radius<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Comparison with Standard Form:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Standard Form: [latex](x-h)^2+(y-k)^2 = r^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Center: [latex](h, k)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Radius: [latex]r[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Conversion Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use completing the square for both x and y terms<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Transforms general form to standard form<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Reveals center and radius<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Importance of Conversion:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Makes graphing easier<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Helps in identifying key circle properties<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Convert the general form equation [latex]x^2 + y^2 - 6x + 4y - 12 = 0[\/latex] to standard form and identify the center and radius, then graph the circle.[reveal-answer q=\"956675\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"956675\"]\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 1:<\/strong> Move the constant term to the other side<\/p>\r\n\r\n<center>[latex]x^2 + y^2 - 6x + 4y = 12[\/latex]<\/center>\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 2:<\/strong> Group the [latex]x[\/latex] and [latex]y[\/latex] terms<\/p>\r\n\r\n<center>[latex](x^2 - 6x) + (y^2 + 4y) = 12[\/latex]<\/center>\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 3:<\/strong> Complete the square for [latex]x[\/latex] and [latex]y[\/latex]<\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">For [latex]x[\/latex]: [latex]x^2 - 6x[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Take half of [latex]-6[\/latex], which is [latex]-3[\/latex], and square it, giving [latex]9[\/latex]. Add and subtract:\r\n<center>[latex]x^2 - 6x + 9 - 9 = (x^2 - 6x + 9) - 9 = (x - 3)^2 - 9[\/latex]<\/center><\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">For [latex]y[\/latex]: [latex]y^2 + 4y[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Take half of [latex]4[\/latex], which is [latex]2[\/latex], and square it, giving [latex]4[\/latex]. Add and subtract:\r\n<center>[latex]y^2 + 4y + 4 - 4 = (y^2 + 4y + 4) - 4 = (y + 2)^2 - 4[\/latex]<\/center><\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 4:<\/strong> Rewrite the equation with the completed squares<\/p>\r\n\r\n<center>[latex](x - 3)^2 - 9 + (y + 2)^2 - 4 = 12[\/latex]<\/center>\r\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 5:<\/strong> Simplify and move the constant terms to the other side<\/p>\r\n\r\n<center>[latex](x - 3)^2 + (y + 2)^2 - 13 = 12[\/latex]<\/center><center>[latex](x - 3)^2 + (y + 2)^2 = 25[\/latex]<\/center>\r\n<p class=\"whitespace-pre-wrap break-words\">Now we have the standard form: [latex](x - 3)^2 + (y + 2)^2 = 25[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Thus, center: [latex](3, -2)[\/latex] and radius: [latex]r = 5[\/latex].<\/p>\r\nGraph:\r\n\r\n[caption id=\"attachment_5246\" align=\"aligncenter\" width=\"790\"]<img class=\"wp-image-5246 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154.png\" alt=\"a coordinate plane with a red circle centered at the point (3, -2)\" width=\"790\" height=\"676\" \/> Circle on a coordinate plane with the center labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h2>Finding the Equation of a Circle<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Standard Form Equation: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">[latex](h, k)[\/latex] is the center<\/li>\r\n \t<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is the radius<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Three Common Scenarios:\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">Known center and radius<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Known center and a point on the circle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Known two points on the circle (often diameter endpoints)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Key Formulas:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Distance Formula: [latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Midpoint Formula: [latex]\\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Scenario Breakdown<\/strong><\/p>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Scenario 1: Known Center and Radius<\/strong><\/p>\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Directly plug into standard form equation<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Scenario 2: Known Center and Point<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use distance formula to find radius<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plug center and calculated radius into standard form<\/li>\r\n<\/ol>\r\n<p class=\"font-600 text-lg font-bold\"><strong>Scenario 3: Two Points on Circle<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Use midpoint formula to find center<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Use distance formula to find radius (center to either point)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Plug center and calculated radius into standard form<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">A circular running track in a park has two markers on opposite sides. The coordinates of these markers are [latex](-2, 5)[\/latex] and [latex](6, -1)[\/latex]. Find the equation of this circular track in standard form.[reveal-answer q=\"173346\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"173346\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Identify the scenario: We have two points on the circle (diameter endpoints).<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the center using the midpoint formula:\r\n[latex]h = \\frac{-2 + 6}{2} = 2[\/latex]\r\n[latex]k = \\frac{5 + (-1)}{2} = 2[\/latex]\r\nCenter: [latex](2, 2)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Calculate the radius using the distance formula:\r\n[latex]r = \\sqrt{(6 - 2)^2 + (-1 - 2)^2} = \\sqrt{4^2 + (-3)^2} = \\sqrt{16 + 9} = \\sqrt{25} = 5[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Write the equation using standard form:\r\n[latex](x - 2)^2 + (y - 2)^2 = 5^2[\/latex]\r\n[latex](x - 2)^2 + (y - 2)^2 = 25[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Verify the solution:\r\nFor [latex](-2, 5)[\/latex]: [latex](-2 - 2)^2 + (5 - 2)^2 = (-4)^2 + 3^2 = 16 + 9 = 25[\/latex] \u2713\r\nFor [latex](6, -1)[\/latex]: [latex](6 - 2)^2 + (-1 - 2)^2 = 4^2 + (-3)^2 = 16 + 9 = 25[\/latex] \u2713<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the equation of the circular running track is [latex](x - 2)^2 + (y - 2)^2 = 25[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bhbhadhe-k5Z3qb052Ek\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/k5Z3qb052Ek?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bhbhadhe-k5Z3qb052Ek\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851179&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bhbhadhe-k5Z3qb052Ek&vembed=0&video_id=k5Z3qb052Ek&video_target=tpm-plugin-bhbhadhe-k5Z3qb052Ek'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+Standard+Equation+of+a+Circle+Given+Center+and+Point+on+the+Circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find Standard Equation of a Circle Given Center and Point on the Circle\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Intersection of a Circle and a Line<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\n<ul>\r\n \t<li class=\"whitespace-normal break-words\">System of Equations:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Circle: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Line: [latex]y = mx + b[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solving Process:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute line equation into circle equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve resulting quadratic equation<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Types of Intersections:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">No intersection: No real solutions<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Tangent: One real solution<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two intersections: Two real solutions<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Geometric Interpretation:\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">No intersection: Line outside circle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">One intersection: Line touches circle at one point<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Two intersections: Line passes through circle<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-600 text-xl font-bold\"><strong>Solution Process<\/strong><\/p>\r\n\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute [latex]y = mx + b[\/latex] into circle equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Simplify to get quadratic in [latex]x[\/latex]: [latex]ax^2 + bx + c = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve quadratic using preferred method (factoring, quadratic formula, etc.)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find corresponding [latex]y[\/latex]-values using line equation<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check solutions in both original equations<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">Find the intersection points of the circle [latex]x^2 + y^2 = 10[\/latex] and the line [latex]y = 2x - 1[\/latex].[reveal-answer q=\"576575\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"576575\"]\r\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\r\n \t<li class=\"whitespace-normal break-words\">Substitute [latex]y = 2x - 1[\/latex] into [latex]x^2 + y^2 = 10[\/latex]:\r\n[latex]x^2 + (2x - 1)^2 = 10[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Expand and simplify:\r\n[latex]x^2 + 4x^2 - 4x + 1 = 10[\/latex]\r\n[latex]5x^2 - 4x - 9 = 0[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Solve the quadratic equation:\r\n[latex]x = \\frac{4 \\pm \\sqrt{16 + 180}}{10} = \\frac{4 \\pm \\sqrt{196}}{10} = \\frac{4 \\pm 14}{10}[\/latex] [latex]x_1 = \\frac{18}{10} = 1.8[\/latex] and [latex]x_2 = -\\frac{10}{10} = -1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find corresponding [latex]y[\/latex]-values:\r\nFor [latex]x_1 = 1.8[\/latex]: [latex]y_1 = 2(1.8) - 1 = 2.6[\/latex]\r\nFor [latex]x_2 = -1[\/latex]: [latex]y_2 = 2(-1) - 1 = -3[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Check solutions (approximate for first point):\r\n[latex](1.8)^2\u00a0+ (2.6)^2 \\approx 10[\/latex] and [latex]2.6 \\approx 2(1.8) - 1[\/latex]\r\n[latex](-1)^2 + (-3)^2 = 10[\/latex] and [latex]-3 = 2(-1) - 1[\/latex]<\/li>\r\n<\/ol>\r\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the intersection points are approximately [latex](1.8, 2.6)[\/latex] and exactly [latex](-1, -3)[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gdcffdcd-kTZhEM4ap-M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kTZhEM4ap-M?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-gdcffdcd-kTZhEM4ap-M\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851180&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gdcffdcd-kTZhEM4ap-M&vembed=0&video_id=kTZhEM4ap-M&video_target=tpm-plugin-gdcffdcd-kTZhEM4ap-M'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Find+a+Point+of+Intersection+of+a+Line+and+a+Circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Find a Point of Intersection of a Line and a Circle\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Write the equations for circles using the standard form<\/li>\n<li>Graph a circle<\/li>\n<li>Solve system of equations involving circles<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Write the Equation of a Circle in Standard Form<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Circle Definition:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Set of points equidistant from a center point in a plane<\/li>\n<li class=\"whitespace-normal break-words\">Center: [latex](h,k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Radius: [latex]r[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Standard Form Equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](x-h)^2+(y-k)^2 = r^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex](h,k)[\/latex] represents the center<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is the radius<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Components of the Equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](x-h)^2[\/latex]: horizontal distance from center<\/li>\n<li class=\"whitespace-normal break-words\">[latex](y-k)^2[\/latex]: vertical distance from center<\/li>\n<li class=\"whitespace-normal break-words\">Right side [latex]r^2[\/latex]: squared radius<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Variations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center at origin: [latex]x^2+y^2 = r^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Negative values inside parentheses change to addition<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Write the equation of a circle with center [latex](-2, 5)[\/latex] and radius [latex]4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q822019\">Show Answer<\/button><\/p>\n<div id=\"q822019\" class=\"hidden-answer\" style=\"display: none\">Identify the values:<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center: [latex]h = -2[\/latex], [latex]k = 5[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Radius: [latex]r = 4[\/latex]<\/li>\n<\/ul>\n<p>Plug these values into the standard form:<\/p>\n<div style=\"text-align: center;\">[latex](x-(-2))^2+(y-(5))^2 = 4^2[\/latex]<\/div>\n<p>Simplify:<\/p>\n<div style=\"text-align: center;\">[latex](x+2)^2+(y-5)^2 = 16[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-aedhdgga-Xmde2JLKAFQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Xmde2JLKAFQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-aedhdgga-Xmde2JLKAFQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12851177&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-aedhdgga-Xmde2JLKAFQ&#38;vembed=0&#38;video_id=Xmde2JLKAFQ&#38;video_target=tpm-plugin-aedhdgga-Xmde2JLKAFQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+1+-+Write+General+Equation+of+a+Circle+in+Standard+Form_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 1: Write General Equation of a Circle in Standard Form\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2 data-type=\"title\">Graph a Circle<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Equation to Graph Relationship:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Standard form: [latex](x-h)^2 + (y-k)^2 = r^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Each part of the equation corresponds to a graphical feature<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Components for Graphing:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center: [latex](h,k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Radius: [latex]r[\/latex] (square root of the right side)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Critical Points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Center: [latex](h,k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Top: [latex](h, k+r)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Bottom: [latex](h, k-r)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Right: [latex](h+r, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Left: [latex](h-r, k)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Symmetry:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Circles are symmetrical about their center<\/li>\n<li class=\"whitespace-normal break-words\">Vertical and horizontal lines through the center are lines of symmetry<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the center and radius, then graph the circle:<\/p>\n<div style=\"text-align: center;\">[latex](x+2)^2+(y\u22121)^2=9[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q738682\">Show Answer<\/button><\/p>\n<div id=\"q738682\" class=\"hidden-answer\" style=\"display: none\">Use the standard form of the equation of a circle. Identify the center, [latex](h,k)[\/latex] and radius, [latex]r[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex](x+2)^2+(y\u22121)^2=9[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex](x-(-2))^2 + (y-(1))^2 = 3^2[\/latex]<\/p>\n<p><strong>Thus, the center is [latex](-2,1)[\/latex] and the radius is [latex]3[\/latex].<\/strong><\/p>\n<p>Graph the circle:<\/p>\n<figure id=\"attachment_5238\" aria-describedby=\"caption-attachment-5238\" style=\"width: 318px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5238 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14134957\/22466db6b6d5f98c6597ff0004e6085ee3b4255b.jpg\" alt=\"a coordinate plane with a circle centered at the point (-2, 1). The circle has a radius of 3, indicated by a labeled line segment from the center to a point on the circle's circumference.\" width=\"318\" height=\"323\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14134957\/22466db6b6d5f98c6597ff0004e6085ee3b4255b.jpg 318w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14134957\/22466db6b6d5f98c6597ff0004e6085ee3b4255b-295x300.jpg 295w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14134957\/22466db6b6d5f98c6597ff0004e6085ee3b4255b-65x66.jpg 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14134957\/22466db6b6d5f98c6597ff0004e6085ee3b4255b-225x229.jpg 225w\" sizes=\"(max-width: 318px) 100vw, 318px\" \/><figcaption id=\"caption-attachment-5238\" class=\"wp-caption-text\">Circle on a coordinate plane with the center and radius labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-pre-wrap break-words\">Graph the circle given by the equation: [latex](x+1)^2 + (y-3)^2 = 9[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q981055\">Show Answer<\/button><\/p>\n<div id=\"q981055\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Identify center and radius:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Rewrite in standard form: [latex](x-(-1))^2 + (y-3)^2 = 3^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Center: [latex](-1,3)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Radius: [latex]r = 3[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Plot the center at [latex](-1, 3)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Mark the four key points:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Top: [latex](-1, 6)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Bottom: [latex](-1, 0)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Right: [latex](2, 3)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Left: [latex](-4, 3)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Connect these points with a smooth circular curve<br \/>\n<figure id=\"attachment_5239\" aria-describedby=\"caption-attachment-5239\" style=\"width: 681px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5239 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14140126\/Screenshot-2024-10-14-100106.png\" alt=\"a coordinate plane with a red circle centered at the point (-1, 3). The radius of the circle is indicated by several labeled points on its circumference: (-1, 6), (-1, 0), (2, 3), and (-4, 3).\" width=\"681\" height=\"637\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14140126\/Screenshot-2024-10-14-100106.png 681w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14140126\/Screenshot-2024-10-14-100106-300x281.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14140126\/Screenshot-2024-10-14-100106-65x61.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14140126\/Screenshot-2024-10-14-100106-225x210.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14140126\/Screenshot-2024-10-14-100106-350x327.png 350w\" sizes=\"(max-width: 681px) 100vw, 681px\" \/><figcaption id=\"caption-attachment-5239\" class=\"wp-caption-text\">Circle on a coordinate plane with the center and four points labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-ddhbcdeb-dm85p_X_L4A\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/dm85p_X_L4A?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-ddhbcdeb-dm85p_X_L4A\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12851178&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-ddhbcdeb-dm85p_X_L4A&#38;vembed=0&#38;video_id=dm85p_X_L4A&#38;video_target=tpm-plugin-ddhbcdeb-dm85p_X_L4A\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Graph+a+Circle+-+Write+the+Equation+in+Standard+form+x%5E2%2By%5E2-10y%2B16%3D0_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cGraph a Circle: Write the Equation in Standard form x^2+y^2-10y+16=0\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>General Form of the Equation of a Circle<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">General Form Equation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]x^2 + y^2 + ax + by + c = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]a[\/latex], [latex]b[\/latex], and [latex]c[\/latex] are constants<\/li>\n<li class=\"whitespace-normal break-words\">Does not directly show center or radius<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Comparison with Standard Form:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Standard Form: [latex](x-h)^2+(y-k)^2 = r^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Center: [latex](h, k)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Radius: [latex]r[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conversion Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use completing the square for both x and y terms<\/li>\n<li class=\"whitespace-normal break-words\">Transforms general form to standard form<\/li>\n<li class=\"whitespace-normal break-words\">Reveals center and radius<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Importance of Conversion:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Makes graphing easier<\/li>\n<li class=\"whitespace-normal break-words\">Helps in identifying key circle properties<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Convert the general form equation [latex]x^2 + y^2 - 6x + 4y - 12 = 0[\/latex] to standard form and identify the center and radius, then graph the circle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q956675\">Show Answer<\/button><\/p>\n<div id=\"q956675\" class=\"hidden-answer\" style=\"display: none\">\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 1:<\/strong> Move the constant term to the other side<\/p>\n<div style=\"text-align: center;\">[latex]x^2 + y^2 - 6x + 4y = 12[\/latex]<\/div>\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 2:<\/strong> Group the [latex]x[\/latex] and [latex]y[\/latex] terms<\/p>\n<div style=\"text-align: center;\">[latex](x^2 - 6x) + (y^2 + 4y) = 12[\/latex]<\/div>\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 3:<\/strong> Complete the square for [latex]x[\/latex] and [latex]y[\/latex]<\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">For [latex]x[\/latex]: [latex]x^2 - 6x[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Take half of [latex]-6[\/latex], which is [latex]-3[\/latex], and square it, giving [latex]9[\/latex]. Add and subtract:\n<div style=\"text-align: center;\">[latex]x^2 - 6x + 9 - 9 = (x^2 - 6x + 9) - 9 = (x - 3)^2 - 9[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">For [latex]y[\/latex]: [latex]y^2 + 4y[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Take half of [latex]4[\/latex], which is [latex]2[\/latex], and square it, giving [latex]4[\/latex]. Add and subtract:\n<div style=\"text-align: center;\">[latex]y^2 + 4y + 4 - 4 = (y^2 + 4y + 4) - 4 = (y + 2)^2 - 4[\/latex]<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 4:<\/strong> Rewrite the equation with the completed squares<\/p>\n<div style=\"text-align: center;\">[latex](x - 3)^2 - 9 + (y + 2)^2 - 4 = 12[\/latex]<\/div>\n<p class=\"whitespace-pre-wrap break-words\"><strong>Step 5:<\/strong> Simplify and move the constant terms to the other side<\/p>\n<div style=\"text-align: center;\">[latex](x - 3)^2 + (y + 2)^2 - 13 = 12[\/latex]<\/div>\n<div style=\"text-align: center;\">[latex](x - 3)^2 + (y + 2)^2 = 25[\/latex]<\/div>\n<p class=\"whitespace-pre-wrap break-words\">Now we have the standard form: [latex](x - 3)^2 + (y + 2)^2 = 25[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Thus, center: [latex](3, -2)[\/latex] and radius: [latex]r = 5[\/latex].<\/p>\n<p>Graph:<\/p>\n<figure id=\"attachment_5246\" aria-describedby=\"caption-attachment-5246\" style=\"width: 790px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-5246 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154.png\" alt=\"a coordinate plane with a red circle centered at the point (3, -2)\" width=\"790\" height=\"676\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154.png 790w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154-300x257.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154-768x657.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154-65x56.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154-225x193.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/42\/2024\/08\/14141211\/Screenshot-2024-10-14-101154-350x299.png 350w\" sizes=\"(max-width: 790px) 100vw, 790px\" \/><figcaption id=\"caption-attachment-5246\" class=\"wp-caption-text\">Circle on a coordinate plane with the center labeled<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<h2>Finding the Equation of a Circle<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Standard Form Equation: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex](h, k)[\/latex] is the center<\/li>\n<li class=\"whitespace-normal break-words\">[latex]r[\/latex] is the radius<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Three Common Scenarios:\n<ul>\n<li class=\"whitespace-normal break-words\">Known center and radius<\/li>\n<li class=\"whitespace-normal break-words\">Known center and a point on the circle<\/li>\n<li class=\"whitespace-normal break-words\">Known two points on the circle (often diameter endpoints)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Distance Formula: [latex]d = \\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Midpoint Formula: [latex]\\left( \\frac{x_1 + x_2}{2}, \\frac{y_1 + y_2}{2} \\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Scenario Breakdown<\/strong><\/p>\n<p class=\"font-600 text-lg font-bold\"><strong>Scenario 1: Known Center and Radius<\/strong><\/p>\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Directly plug into standard form equation<\/li>\n<\/ul>\n<p class=\"font-600 text-lg font-bold\"><strong>Scenario 2: Known Center and Point<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use distance formula to find radius<\/li>\n<li class=\"whitespace-normal break-words\">Plug center and calculated radius into standard form<\/li>\n<\/ol>\n<p class=\"font-600 text-lg font-bold\"><strong>Scenario 3: Two Points on Circle<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Use midpoint formula to find center<\/li>\n<li class=\"whitespace-normal break-words\">Use distance formula to find radius (center to either point)<\/li>\n<li class=\"whitespace-normal break-words\">Plug center and calculated radius into standard form<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">A circular running track in a park has two markers on opposite sides. The coordinates of these markers are [latex](-2, 5)[\/latex] and [latex](6, -1)[\/latex]. Find the equation of this circular track in standard form.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q173346\">Show Answer<\/button><\/p>\n<div id=\"q173346\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Identify the scenario: We have two points on the circle (diameter endpoints).<\/li>\n<li class=\"whitespace-normal break-words\">Find the center using the midpoint formula:<br \/>\n[latex]h = \\frac{-2 + 6}{2} = 2[\/latex]<br \/>\n[latex]k = \\frac{5 + (-1)}{2} = 2[\/latex]<br \/>\nCenter: [latex](2, 2)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Calculate the radius using the distance formula:<br \/>\n[latex]r = \\sqrt{(6 - 2)^2 + (-1 - 2)^2} = \\sqrt{4^2 + (-3)^2} = \\sqrt{16 + 9} = \\sqrt{25} = 5[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Write the equation using standard form:<br \/>\n[latex](x - 2)^2 + (y - 2)^2 = 5^2[\/latex]<br \/>\n[latex](x - 2)^2 + (y - 2)^2 = 25[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Verify the solution:<br \/>\nFor [latex](-2, 5)[\/latex]: [latex](-2 - 2)^2 + (5 - 2)^2 = (-4)^2 + 3^2 = 16 + 9 = 25[\/latex] \u2713<br \/>\nFor [latex](6, -1)[\/latex]: [latex](6 - 2)^2 + (-1 - 2)^2 = 4^2 + (-3)^2 = 16 + 9 = 25[\/latex] \u2713<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the equation of the circular running track is [latex](x - 2)^2 + (y - 2)^2 = 25[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bhbhadhe-k5Z3qb052Ek\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/k5Z3qb052Ek?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bhbhadhe-k5Z3qb052Ek\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12851179&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bhbhadhe-k5Z3qb052Ek&#38;vembed=0&#38;video_id=k5Z3qb052Ek&#38;video_target=tpm-plugin-bhbhadhe-k5Z3qb052Ek\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+-+Find+Standard+Equation+of+a+Circle+Given+Center+and+Point+on+the+Circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx: Find Standard Equation of a Circle Given Center and Point on the Circle\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Intersection of a Circle and a Line<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">System of Equations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Circle: [latex](x - h)^2 + (y - k)^2 = r^2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Line: [latex]y = mx + b[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Solving Process:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute line equation into circle equation<\/li>\n<li class=\"whitespace-normal break-words\">Solve resulting quadratic equation<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Types of Intersections:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">No intersection: No real solutions<\/li>\n<li class=\"whitespace-normal break-words\">Tangent: One real solution<\/li>\n<li class=\"whitespace-normal break-words\">Two intersections: Two real solutions<\/li>\n<\/ul>\n<\/li>\n<li>Geometric Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">No intersection: Line outside circle<\/li>\n<li class=\"whitespace-normal break-words\">One intersection: Line touches circle at one point<\/li>\n<li class=\"whitespace-normal break-words\">Two intersections: Line passes through circle<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-600 text-xl font-bold\"><strong>Solution Process<\/strong><\/p>\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute [latex]y = mx + b[\/latex] into circle equation<\/li>\n<li class=\"whitespace-normal break-words\">Simplify to get quadratic in [latex]x[\/latex]: [latex]ax^2 + bx + c = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve quadratic using preferred method (factoring, quadratic formula, etc.)<\/li>\n<li class=\"whitespace-normal break-words\">Find corresponding [latex]y[\/latex]-values using line equation<\/li>\n<li class=\"whitespace-normal break-words\">Check solutions in both original equations<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">Find the intersection points of the circle [latex]x^2 + y^2 = 10[\/latex] and the line [latex]y = 2x - 1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q576575\">Show Answer<\/button><\/p>\n<div id=\"q576575\" class=\"hidden-answer\" style=\"display: none\">\n<ol class=\"-mt-1 list-decimal space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Substitute [latex]y = 2x - 1[\/latex] into [latex]x^2 + y^2 = 10[\/latex]:<br \/>\n[latex]x^2 + (2x - 1)^2 = 10[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Expand and simplify:<br \/>\n[latex]x^2 + 4x^2 - 4x + 1 = 10[\/latex]<br \/>\n[latex]5x^2 - 4x - 9 = 0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Solve the quadratic equation:<br \/>\n[latex]x = \\frac{4 \\pm \\sqrt{16 + 180}}{10} = \\frac{4 \\pm \\sqrt{196}}{10} = \\frac{4 \\pm 14}{10}[\/latex] [latex]x_1 = \\frac{18}{10} = 1.8[\/latex] and [latex]x_2 = -\\frac{10}{10} = -1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Find corresponding [latex]y[\/latex]-values:<br \/>\nFor [latex]x_1 = 1.8[\/latex]: [latex]y_1 = 2(1.8) - 1 = 2.6[\/latex]<br \/>\nFor [latex]x_2 = -1[\/latex]: [latex]y_2 = 2(-1) - 1 = -3[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Check solutions (approximate for first point):<br \/>\n[latex](1.8)^2\u00a0+ (2.6)^2 \\approx 10[\/latex] and [latex]2.6 \\approx 2(1.8) - 1[\/latex]<br \/>\n[latex](-1)^2 + (-3)^2 = 10[\/latex] and [latex]-3 = 2(-1) - 1[\/latex]<\/li>\n<\/ol>\n<p class=\"whitespace-pre-wrap break-words\">Therefore, the intersection points are approximately [latex](1.8, 2.6)[\/latex] and exactly [latex](-1, -3)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\" aria-label=\"Watch It\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-gdcffdcd-kTZhEM4ap-M\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/kTZhEM4ap-M?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-gdcffdcd-kTZhEM4ap-M\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12851180&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-gdcffdcd-kTZhEM4ap-M&#38;vembed=0&#38;video_id=kTZhEM4ap-M&#38;video_target=tpm-plugin-gdcffdcd-kTZhEM4ap-M\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Ex+2+-+Find+a+Point+of+Intersection+of+a+Line+and+a+Circle_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEx 2: Find a Point of Intersection of a Line and a Circle\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":12,"menu_order":9,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Ex 1: Write General Equation of a Circle in Standard Form\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/Xmde2JLKAFQ\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Graph a Circle: Write the Equation in Standard form x^2+y^2-10y+16=0\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/dm85p_X_L4A\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex: Find Standard Equation of a Circle Given Center and Point on the Circle\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/k5Z3qb052Ek\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Ex 2: Find a Point of Intersection of a Line and a Circle\",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/youtu.be\/kTZhEM4ap-M\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":345,"module-header":"fresh_take","content_attributions":[{"type":"copyrighted_video","description":"Ex 1: Write General Equation of a Circle in Standard Form","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/Xmde2JLKAFQ","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Graph a Circle: Write the Equation in Standard form x^2+y^2-10y+16=0","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/dm85p_X_L4A","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex: Find Standard Equation of a Circle Given Center and Point on the Circle","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/k5Z3qb052Ek","project":"","license":"arr","license_terms":"Standard YouTube License"},{"type":"copyrighted_video","description":"Ex 2: Find a Point of Intersection of a Line and a Circle","author":"","organization":"Mathispower4u","url":"https:\/\/youtu.be\/kTZhEM4ap-M","project":"","license":"arr","license_terms":"Standard YouTube License"}],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851177&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-aedhdgga-Xmde2JLKAFQ&vembed=0&video_id=Xmde2JLKAFQ&video_target=tpm-plugin-aedhdgga-Xmde2JLKAFQ'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851178&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-ddhbcdeb-dm85p_X_L4A&vembed=0&video_id=dm85p_X_L4A&video_target=tpm-plugin-ddhbcdeb-dm85p_X_L4A'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851179&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bhbhadhe-k5Z3qb052Ek&vembed=0&video_id=k5Z3qb052Ek&video_target=tpm-plugin-bhbhadhe-k5Z3qb052Ek'><\/script>\n<script type='text\/javascript' src='https:\/\/www.youtube.com\/iframe_api'><\/script><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12851180&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-gdcffdcd-kTZhEM4ap-M&vembed=0&video_id=kTZhEM4ap-M&video_target=tpm-plugin-gdcffdcd-kTZhEM4ap-M'><\/script>\n","media_targets":["tpm-plugin-aedhdgga-Xmde2JLKAFQ","tpm-plugin-ddhbcdeb-dm85p_X_L4A","tpm-plugin-bhbhadhe-k5Z3qb052Ek","tpm-plugin-gdcffdcd-kTZhEM4ap-M"]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2555"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":12,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2555\/revisions"}],"predecessor-version":[{"id":7883,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2555\/revisions\/7883"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/345"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/2555\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=2555"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=2555"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=2555"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=2555"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}