{"id":9145,"date":"2020-12-09T12:06:16","date_gmt":"2020-12-09T06:36:16","guid":{"rendered":"https:\/\/cbselibrary.com\/?p=9145"},"modified":"2020-12-09T15:30:03","modified_gmt":"2020-12-09T10:00:03","slug":"equation-of-circles","status":"publish","type":"post","link":"https:\/\/cbselibrary.com\/equation-of-circles\/","title":{"rendered":"Equation of Circles"},"content":{"rendered":"<h2><span style=\"color: #3366ff;\"><strong>Equation of Circles<\/strong><\/span><\/h2>\n<p><iframe loading=\"lazy\" title=\"Equation of a circle 1\" width=\"1200\" height=\"675\" src=\"https:\/\/www.youtube.com\/embed\/FjYaBqbCjbQ?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen><\/iframe><\/p>\n<p>Let&#8217;s review what we already know about circles.<br \/>\n<strong>Definition:<\/strong> A circle is a locus (set) of points in a plane equidistant from a fixed point.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128309\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-1.png\" alt=\"Equation of Circles 1\" width=\"635\" height=\"439\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-1.png 635w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-1-300x207.png 300w\" sizes=\"(max-width: 635px) 100vw, 635px\" \/><\/p>\n<p>Now, if we &#8220;multiply out&#8221; the above example (x-2)<sup>2 <\/sup>+ (y+5)<sup>2\u00a0 <\/sup>= 9 we will get:<\/p>\n<p>(x-2)<sup>2 <\/sup>+ (y+5)<sup>2\u00a0 <\/sup>= 9<br \/>\n(x<sup>2<\/sup>-4x+4) +( y<sup>2<\/sup>+10y+25) = 9<br \/>\nx<sup>2<\/sup>-4x+4+ y<sup>2<\/sup>+10y+25 = 9<br \/>\nx<sup>2<\/sup>+ y<sup>2<\/sup>-4x+10y+20= 0<br \/>\nx<sup>2<\/sup>+y<sup>2<\/sup>+Cx+Dy+E = 0<\/p>\n<p>When multiplied out, we obtain the &#8220;general form&#8221; of the equation of a circle. Notice that in this form we can clearly see that the equation of a circle has both x<sup>2<\/sup>\u00a0and y<sup>2<\/sup> terms and these terms have the same coefficient (usually 1).<\/p>\n<p>When the equation of a circle appears in &#8220;general form&#8221;, it is often beneficial to convert the equation to &#8220;center-radius&#8221; form to easily read the center coordinates and the radius for graphing.<br \/>\n<img loading=\"lazy\" class=\"alignnone  wp-image-128311\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-2.jpg\" alt=\"Equation of Circles 2\" width=\"666\" height=\"500\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-2.jpg 960w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-2-300x225.jpg 300w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-2-768x576.jpg 768w\" sizes=\"(max-width: 666px) 100vw, 666px\" \/><\/p>\n<p><strong>Examples:<\/strong><\/p>\n<p><strong>1. Convert x<sup>2<\/sup>+ y<sup>2<\/sup>-4x-6y+8= 0\u00a0 into center-radius form.<\/strong><\/p>\n<p>This conversion requires use of the technique of completing the square.<\/p>\n<p>We will be creating two perfect square trinomials within the equation.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128312\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-3.png\" alt=\"Equation of Circles 3\" width=\"328\" height=\"160\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-3.png 328w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-3-300x146.png 300w\" sizes=\"(max-width: 328px) 100vw, 328px\" \/><\/p>\n<ul>\n<li>Start by grouping the x related terms together and the y related terms together. Move any numerical constants (plain numbers) to the other side.<\/li>\n<li>Get ready to insert the needed values for creating the perfect square trinomials. Remember to balance both sides of the equation.<\/li>\n<li>Find each missing value by taking half of the &#8220;middle term&#8221; and squaring. This value will always be positive as a result of the squaring process.<\/li>\n<li>Rewrite in factored form.<br \/>\nYou can now read that the center of the circle is at (2, 3) and the radius is \/5<\/li>\n<\/ul>\n<p><strong>2. How do the coordinates of the center of a circle relate to C and D when the equation of the circle is in the general form x<sup>2<\/sup>+ y<sup>2<\/sup>+Cx+Dy+E = 0.<\/strong><\/p>\n<p>Let&#8217;s make some observations. Re-examine our previous equations in general form and center-radius form. Do you see a relationship between the center coordinates and C and D?<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128313\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-4.png\" alt=\"Equation of Circles 4\" width=\"481\" height=\"128\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-4.png 481w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-4-300x80.png 300w\" sizes=\"(max-width: 481px) 100vw, 481px\" \/><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128314\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-5.png\" alt=\"Equation of Circles 5\" width=\"631\" height=\"119\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-5.png 631w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-5-300x57.png 300w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-5-630x119.png 630w\" sizes=\"(max-width: 631px) 100vw, 631px\" \/><\/p>\n<p><strong>3. Write the equation of a circle whose diameter has endpoints (4, -1) and (-6, 7).<\/strong><\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128315\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-6.png\" alt=\"Equation of Circles 6\" width=\"637\" height=\"292\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-6.png 637w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-6-300x138.png 300w\" sizes=\"(max-width: 637px) 100vw, 637px\" \/><\/p>\n<p><strong>4. Write the equation for the circle shown below if it is shifted 3 units to the right and 4 units up.<\/strong><\/p>\n<p>A shift of 3 units to the right and 4 units up places the center at the point (3, 4). The radius of the circle can be seen from the graph to be 5.<br \/>\nEquation:(x-3)<sup>2 <\/sup>+ (y-4)<sup>2\u00a0 <\/sup>= 25<\/p>\n<p><strong>5. Convert 2x<sup>2<\/sup>+ 2y<sup>2<\/sup>+6x-8y+12= 0 into center-radius form.<\/strong><\/p>\n<p>Whoa!!! This equation looks different. Are we sure this is a circle???<br \/>\nIn this equation, both the x and y terms appear in squared form and their coefficients (the numbers in front of them) are the same. Yes, we have a circle here! We will, however, have to deal with the coefficients of 2 before we can complete the square.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-128316\" src=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-7.png\" alt=\"Equation of Circles 7\" width=\"593\" height=\"252\" srcset=\"https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-7.png 593w, https:\/\/cbselibrary.com\/wp-content\/uploads\/2017\/04\/Equation-of-Circles-7-300x127.png 300w\" sizes=\"(max-width: 593px) 100vw, 593px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Equation of Circles Let&#8217;s review what we already know about circles. Definition: A circle is a locus (set) of points in a plane equidistant from a fixed point. Now, if we &#8220;multiply out&#8221; the above example (x-2)2 + (y+5)2\u00a0 = 9 we will get: (x-2)2 + (y+5)2\u00a0 = 9 (x2-4x+4) +( y2+10y+25) = 9 x2-4x+4+ &#8230; <a title=\"Equation of Circles\" class=\"read-more\" href=\"https:\/\/cbselibrary.com\/equation-of-circles\/\" aria-label=\"More on Equation of Circles\">Read more<\/a><\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"spay_email":""},"categories":[5],"tags":[3351],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v17.8 (Yoast SEO v18.4.1) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Equation of Circles - CBSE Library<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/cbselibrary.com\/equation-of-circles\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Circles\" \/>\n<meta property=\"og:description\" content=\"Equation of Circles Let&#8217;s review what we already know about circles. Definition: A circle is a locus (set) of points in a plane equidistant from a fixed point. Now, if we &#8220;multiply out&#8221; the above example (x-2)2 + (y+5)2\u00a0 = 9 we will get: (x-2)2 + (y+5)2\u00a0 = 9 (x2-4x+4) +( y2+10y+25) = 9 x2-4x+4+ ... 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Definition: A circle is a locus (set) of points in a plane equidistant from a fixed point. Now, if we &#8220;multiply out&#8221; the above example (x-2)2 + (y+5)2\u00a0 = 9 we will get: (x-2)2 + (y+5)2\u00a0 = 9 (x2-4x+4) +( y2+10y+25) = 9 x2-4x+4+ ... 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