{"id":3884,"date":"2021-08-09T19:50:41","date_gmt":"2021-08-09T19:50:41","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3884"},"modified":"2021-11-20T22:43:07","modified_gmt":"2021-11-20T17:13:07","slug":"common-tangent-to-two-circles","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/common-tangent-to-two-circles\/","title":{"rendered":"Common Tangent to Two Circles \u2013 Direct and Transverse"},"content":{"rendered":"

Here you will learn common tangent to two circles i.e direct and transverse common tangents to two circles.<\/p>\n

Let’s begin –<\/p>\n

Common Tangent to Two Circles<\/h2>\n

Let two circles having centers \\(C_1\\) and \\(C_2\\) and radii, \\(r_1\\) and \\(r_2\\) and \\(C_1\\)\\(C_2\\) is the distance between their centres then :<\/p>\n

(a)  Both circles will touch :<\/strong><\/p>\n

(i)  Externally : <\/strong>if \\(C_1\\)\\(C_2\\) = \\(r_1\\)\\(r_2\\) i.e, the distance between their centres is equal to sum of their radii and point P & T divides \\(C_1\\)\\(C_2\\) in the ratio \\(r_1\\) : \\(r_2\\) (internally & externally respectively). In this case there are three common tangents<\/strong>.<\/p>\n

(ii) Internally : <\/strong>if \\(C_1\\)\\(C_2\\) = |\\(r_1\\) – \\(r_2\\)| i.e, the distance between their centres is equal to difference between their radii and point P divides \\(C_1\\)\\(C_2\\) in the ratio \\(r_1\\) : \\(r_2\\) externally and in this case there will be only <b>one common tangent.<\/p>\n

(b) The circles will intersect :<\/strong><\/p>\n

when |\\(r_1\\) – \\(r_2\\)| < \\(C_1\\)\\(C_2\\) < \\(r_1\\) + \\(r_2\\) in this case there are two common tangent<\/strong>.<\/p>\n

(c) The circles will not intersect :<\/strong><\/p>\n

(i)  One Circle will lie inside the other circle if \\(C_1\\)\\(C_2\\) < |\\(r_1\\) – \\(r_2\\)| In this case there will be no common tangent.<\/strong><\/p>\n

(ii)  When circle are apart from each other then \\(C_1\\)\\(C_2\\) > \\(r_1\\) + \\(r_2\\) and in this case there will be four common tangents.<\/strong><\/p>\n

Direct and Transverse Common Tangents<\/h2>\n

Let two circles having centers C1 and C2 and radii, r1 and r2 and C1C2 is the distance between their centres.\"common<\/p>\n

Lines PQ and Rs are called transverse<\/strong> or indirect<\/strong> or internal<\/strong> common tangents and these lines meet line C1C2 on T1 and T2 divide the line C1C2 in the ratio r1 : r2 internally and lines AB & CD are called direct<\/strong> or external<\/strong> common tangents. These lines meet C1C2 produced on T2. Thus T2 divides C1C2 externally in the ratio r1 : r2.<\/p>\n

Length of Direct Common Tangent<\/h3>\n
\n

Length = \\(\\sqrt{(C1C2)^2 – (r1 – r2)^2}\\)<\/p>\n<\/blockquote>\n

Length of Transverse Common Tangent<\/h3>\n
\n

Length = \\(\\sqrt{(C1C2)^2 – (r1 + r2)^2}\\)<\/p>\n<\/blockquote>\n

Example<\/span><\/strong> : Find the number of common tangents to the circles \\(x^2 + y^2\\) = 1 and \\(x^2 + y^2 – 2x – 6y + 6\\) = 0<\/p>\n

Solution<\/span><\/strong> : Let \\(C_1\\) be the center of circle \\(x^2 + y^2\\) = 1 i.e.  \\(C_1\\) = (0, 0)<\/p>\n

And \\(C_2\\) be the center of circle \\(x^2 + y^2 – 2x – 6y + 6\\) = 0 i.e. \\(C_2\\) = (1, 3)<\/p>\n

Let \\(r_1\\) be the radius of first circle and \\(r_2\\) be the radius of second circle.<\/p>\n

Then, \\(r_1\\) = 1 and \\(r_2\\) = 2<\/p>\n

Learn how to find center and radius of circle here<\/a>. <\/p>\n

Now, \\(C_1C_2\\) = \\(\\sqrt{9 + 1}\\) = \\(\\sqrt{10}\\)  and \\(r_1 + r_2\\) = 3<\/p>\n

Since, \\(C_1C_2\\) > \\(r_1 + r_2\\) <\/p>\n

Hence, there are four common tangents.<\/p>\n\n\n

\n
Next – Equation of Family of Circles<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Equation of Normal to a Circle with Examples<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn common tangent to two circles i.e direct and transverse common tangents to two circles. Let’s begin – Common Tangent to Two Circles Let two circles having centers \\(C_1\\) and \\(C_2\\) and radii, \\(r_1\\) and \\(r_2\\) and \\(C_1\\)\\(C_2\\) is the distance between their centres then : (a)  Both circles will touch : …<\/p>\n

Common Tangent to Two Circles \u2013 Direct and Transverse<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[23],"tags":[181,182,184,183],"yoast_head":"\nCommon Tangent to Two Circles \u2013 Direct and Transverse<\/title>\n<meta name=\"description\" content=\"In this post, you will learn common tangent to two circles i.e direct and transverse common 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