{"id":144,"date":"2017-08-23T09:09:40","date_gmt":"2017-08-23T09:09:40","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=144"},"modified":"2017-09-18T18:47:30","modified_gmt":"2017-09-18T18:47:30","slug":"the-equation-of-a-circle","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/the-equation-of-a-circle\/","title":{"rendered":"The Equation of a Circle"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/circles\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometric-proofs-using-coordinate-systems\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">The Equation of a Circle<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will determine the equation of a circle in the plane centered at a given point (h, k) with a\u00a0given radius r. Also, given an equation of the circle, you will show how to find its center and radius.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li><strong>Pythagorean Theorem : <\/strong><span style=\"text-decoration: none;\">The\u00a0square of the hypotenuse is equal to the sum of the squares of\u00a0the non-hypotenuse legs in any right triangle. <\/span><\/li>\n<li><span style=\"text-decoration: none;\">Let <\/span><em><span style=\"text-decoration: none;\">P\u00a0<\/span><\/em><span style=\"text-decoration: none;\">be a point in the plane and let <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">be a positive real number. The <\/span><strong><em><span style=\"text-decoration: none;\">circle\u00a0<\/span><\/em><\/strong><span style=\"text-decoration: none;\">with\u00a0<\/span><em><strong><span style=\"text-decoration: none;\">center <\/span><\/strong><span style=\"text-decoration: none;\">P\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and radius <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is the set of all points in the plane with distance <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">from <\/span><em><span style=\"text-decoration: none;\">P<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/li>\n<\/ul>\n<section>\n<h3><strong>How do we find the equation of a circle in the plane,\u00a0given its center and radius?<\/strong><\/h3>\n<p><span style=\"text-decoration: none;\">Before answering this\u00a0question generally, consider a special case. Suppose we have a\u00a0circle of radius <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">centered at the origin. Let (<\/span><em><span style=\"text-decoration: none;\">x,\u00a0y<\/span><\/em><span style=\"text-decoration: none;\">) be any point of\u00a0the circle. The distance between (<\/span><em><span style=\"text-decoration: none;\">x,\u00a0y<\/span><\/em><span style=\"text-decoration: none;\">) and (0, 0) is\u00a0<\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">,\u00a0by definition of the circle. If (<\/span><em><span style=\"text-decoration: none;\">x,\u00a0y<\/span><\/em><span style=\"text-decoration: none;\">) is not one of\u00a0the four special points (<\/span><em><span style=\"text-decoration: none;\">r,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">0), (<\/span><em><span style=\"text-decoration: none;\">\u2013r,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">0), (0, <\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">),\u00a0(0, \u2013<\/span><em><span style=\"text-decoration: none;\">r<\/span><\/em><span style=\"text-decoration: none;\">),\u00a0then we can always draw a right triangle with hypotenuse length <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and legs of length <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">as shown below. <\/span><\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.4%20Art%20017.JPG\" alt=\" Circle with inscribed right triangle\" width=\"218\" height=\"218\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" hspace=\"5\" \/><\/em><\/p>\n<p><span style=\"text-decoration: none;\">So, by the Pythagorean\u00a0Theorem, <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <\/span><em><span style=\"text-decoration: none;\">y\u00a0<\/span><\/em><span style=\"text-decoration: none;\">satisfy the equation <\/span><em><span style=\"text-decoration: none;\">x<sup>2\u00a0<\/sup>+ y<sup>2<\/sup> = r<sup>2\u00a0<\/sup><\/span><\/em><span style=\"text-decoration: none;\">. (Notice that the four special points satisfy this equation.)\u00a0This is the equation of the circle of radius <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">centered at the origin. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">Now, we can move the circle\u00a0of radius <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">centered at the origin to any point <\/span><em><span style=\"text-decoration: none;\">(p,\u00a0q)<\/span><\/em><span style=\"text-decoration: none;\"> of the plane by\u00a0two graphing transformations: a horizontal shift of <\/span><em><span style=\"text-decoration: none;\">x\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/arrow.gif\" alt=\"arrow\" width=\"20\" height=\"14\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0x \u2013 h<\/span><\/em><span style=\"text-decoration: none;\"> and a\u00a0vertical shift of <\/span><em><span style=\"text-decoration: none;\">y\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/arrow.gif\" width=\"20\" height=\"14\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0y \u2013 k<\/span><\/em><span style=\"text-decoration: none;\">. This\u00a0gives us the equation (<\/span><em><span style=\"text-decoration: none;\">x\u00a0\u2013 h<\/span><\/em><span style=\"text-decoration: none;\">)<sup>2\u00a0<\/sup><\/span><em><span style=\"text-decoration: none;\">+ <\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">y\u00a0\u2013 k<\/span><\/em><span style=\"text-decoration: none;\">)<\/span><em><span style=\"text-decoration: none;\"><sup>2\u00a0<\/sup>= r <\/span><\/em><span style=\"text-decoration: none;\"><sup>2<\/sup>.\u00a0We have shown the following: <\/span><\/p>\n<p><em><span style=\"text-decoration: none;\">The equation\u00a0of a circle in the plane with center <\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">h,\u00a0k<\/span><\/em><span style=\"text-decoration: none;\">)\u00a0<\/span><em><span style=\"text-decoration: none;\">and with radius r is:\u00a0<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">x\u00a0\u2013 h<\/span><\/em><span style=\"text-decoration: none;\">)<sup>2 <\/sup><\/span><em><span style=\"text-decoration: none;\">+\u00a0<\/span><\/em><span style=\"text-decoration: none;\">(<\/span><em><span style=\"text-decoration: none;\">y\u00a0\u2013 k<\/span><\/em><span style=\"text-decoration: none;\">)<sup>2 <\/sup>=\u00a0<\/span><em><span style=\"text-decoration: none;\">r <\/span><\/em><span style=\"text-decoration: none;\"><sup>2<\/sup>.<\/span><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the equation of the circle shown below?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/Math%20Mod%204.4%20Art%20018.JPG\" alt=\"Circle on coordinate plane\" width=\"218\" height=\"218\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p2_clip_image002.gif\" width=\"109\" height=\"17\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p2_clip_image004.gif\" width=\"109\" height=\"17\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<li><em><img loading=\"lazy\" decoding=\"async\" src=\"4\/images\/s14_p2_clip_image006.gif\" width=\"106\" height=\"17\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/em><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p2_clip_image008.gif\" width=\"102\" height=\"17\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is A . The center is (\u20133, 0) and the\u00a0radius is 4 . We use the form\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p2_clip_image010.gif\" width=\"142\" height=\"17\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>. This gives\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p2_clip_image012.gif\" width=\"161\" height=\"17\" name=\"graphics12\" align=\"BOTTOM\" border=\"0\" \/>or\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p2_clip_image013.gif\" width=\"109\" height=\"17\" name=\"graphics13\" align=\"BOTTOM\" border=\"0\" \/><em>.<\/em><\/p>\n<\/div>\n<\/section>\n<p>If we expand this equation, we get another form for the\u00a0equation of the circle called the <abbr title=\" A polynomial with the degree of each term written in descending order. \">standard form<\/abbr>.<\/p>\n<p>The standard form of the equation of a circle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p2_clip_image015.gif\" width=\"166\" height=\"17\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/><em>.<\/em><\/p>\n<h3><strong>Given the equation of a circle in its general form, how\u00a0can we find the center and radius?<\/strong><\/h3>\n<p>To find this information, we can reverse the process of\u00a0expanding the center-radius form of the circle. This reverse\u00a0process is actually the method of completing the square. Recall\u00a0that\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image002.gif\" width=\"148\" height=\"17\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>. If we <span style=\"text-decoration: none;\">have the <\/span><em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\"><sup>2\u00a0<\/sup>and <\/span><em><span style=\"text-decoration: none;\">x <\/span><\/em><span style=\"text-decoration: none;\">terms,\u00a0then to complete the square we need to add a constant term. But if\u00a0we add a term, we need to add it to both sides of the equation to\u00a0maintain equality. <\/span><\/p>\n<p><span style=\"text-decoration: none;\">In the case of a circle, we\u00a0actually need to do complete the square twice, once for <\/span><em><span style=\"text-decoration: none;\">x\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and once for <\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p style=\"text-decoration: none;\" align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s14_p3_html_m21d39007.gif\" width=\"393\" height=\"131\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>So the center of the circle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s14_p3_html_m619ef04f.gif\" width=\"125\" height=\"45\" name=\"graphics5\" align=\"absmiddle\" border=\"0\" \/>\u00a0and the radius\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/s14_p3_html_1bcd4d50.gif\" width=\"165\" height=\"53\" name=\"graphics6\" align=\"absmiddle\" border=\"0\" \/>.<\/p>\n<p>Let&#8217;s try an example. What is the center of the circle given by\u00a0the equation: <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image010.gif\" width=\"163\" height=\"17\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<p><strong>Step 1:<\/strong> Complete the square for the variable\u00a0<em><span style=\"text-decoration: none;\">x<\/span><\/em><span style=\"text-decoration: none;\">.<br \/>\n<\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image012.gif\" width=\"244\" height=\"146\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p><strong>Step 2:<\/strong> Complete the square for the variab<span style=\"text-decoration: none;\">le\u00a0<\/span><em><span style=\"text-decoration: none;\">y<\/span><\/em><span style=\"text-decoration: none;\">. <\/span><\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image014.gif\" width=\"240\" height=\"146\" name=\"graphics9\" align=\"BOTTOM\" border=\"0\" \/><\/em><\/p>\n<p align=\"LEFT\"><strong>Step 3:<\/strong> Take the square root of the\u00a0remaining term to get the radius.<\/p>\n<p align=\"CENTER\"><em><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image016.gif\" width=\"163\" height=\"18\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/em><\/p>\n<p>The center is (2, \u20133) and the r<span style=\"text-decoration: none;\">adius\u00a0is \u00a0<\/span><em><span style=\"text-decoration: none;\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image018.gif\" width=\"18\" height=\"18\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/>.<br \/>\n<\/span><\/em><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the radius of the circle given by the equation\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image020.gif\" width=\"147\" height=\"17\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>?<\/p>\n<ol>\n<li>1<\/li>\n<li>4<\/li>\n<li>3<\/li>\n<li>9<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct answer is C . Complete the square as in the\u00a0preceding example.<\/p>\n<p align=\"CENTER\"><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image022(b).gif\" \/><\/p>\n<p><span style=\"text-decoration: none;\">Because the equation of a\u00a0circle with center (<em>h<\/em>, <em>k<\/em>) and radius <\/span><em><span style=\"text-decoration: none;\">r\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/4\/images\/s14_p3_clip_image024.gif\" width=\"142\" height=\"17\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0<em>r<\/em> = 3<\/span>.<\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/circles\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometry-spatial-reasoning\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/geometric-proofs-using-coordinate-systems\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 The Equation of a Circle Objective In this lesson, you will determine the equation of a circle in the plane centered at a given point (h, k) with a\u00a0given radius r. Also, given an equation of the circle, you will show how to find its center and radius. Previously [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-144","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/144","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=144"}],"version-history":[{"count":8,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/144\/revisions"}],"predecessor-version":[{"id":764,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/144\/revisions\/764"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}