{"id":3875,"date":"2021-08-09T19:53:04","date_gmt":"2021-08-09T19:53:04","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3875"},"modified":"2021-11-20T16:26:47","modified_gmt":"2021-11-20T10:56:47","slug":"family-of-circles","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/family-of-circles\/","title":{"rendered":"Family of Circles – Equations & Examples"},"content":{"rendered":"

Here you will learn equation of family of circles with examples.<\/p>\n

Let’ begin –<\/p>\n

Family of Circles<\/h2>\n

(a)<\/strong>\u00a0 The equation of the family of circles passing through the point of intersection of two circles \\(S_1\\) = 0 & \\(S_2\\) = 0 is :<\/p>\n

\n

\\(S_1\\) + K\\(S_2\\) = 0\u00a0 \u00a0 \u00a0 (K \\(\\ne\\) -1)<\/strong><\/p>\n<\/blockquote>\n

(b)<\/strong>\u00a0 The equation of the family of circles passing through the point of intersection of circle S = 0 & a line L = 0 is given by<\/p>\n

\n

S + KL = 0.<\/strong><\/p>\n<\/blockquote>\n

(c)\u00a0<\/strong> The equation of the family of circles passing through two given points (\\(x_1, y_1\\)) & (\\(x_2, y_2\\)) can be written in the form :<\/p>\n\n\n

(x – \\(x_1\\))(x – \\(x_2\\)) + (y – \\(y_1\\))(y – \\(y_2\\)) + K\\begin{vmatrix}\n\t\t x & y & 1 \\\\ \n\t\t\tx_1 & y_1 & 1 \\\\\n\t\t\tx_2 & y_2 & 1\n\t\t\t\\end{vmatrix}<\/span> = 0 where K is a parameter.

<\/p>\n\n\n

(d)\u00a0<\/strong> The equation of the family of circles touching a fixed line y – \\(y_1\\) = m(x – \\(x_1\\)) at the fixed point (\\(x_1, y_1\\)) is \\({(x – x_1)}^2\\) + \\({(y – y_1)}^2\\) + K [y – \\(y_1\\) – m(x – \\(x_1\\))] = 0,where K is a parameter.<\/p>\n

(e)<\/strong>\u00a0 Family of circles circumscribing a triangle whose sides are given by \\(L_1\\) = 0 ; \\(L_2\\) = 0 & \\(L_3\\) = 0 is given by : \\(L_1\\)\\(L_2\\) + \\(\\lambda\\)\\(L_2\\)\\(L_3\\) + \\(\\mu\\)\u00a0 \\(L_3\\)\\(L_1\\) = 0 provided coefficients of xy = 0 & coefficient of \\(x^2\\) = coefficient of \\(y^2\\).<\/p>\n

(f)<\/strong>\u00a0 Family of circles circumscribing a quadrilateral whose sides are given by \\(L_1\\) = 0, \\(L_2\\) = 0, \\(L_3\\) = 0 & \\(L_4\\) = 0 is \\(L_1\\)\\(L_3\\) + \\(\\lambda\\)\\(L_2\\)\\(L_4\\) provided coefficients of xy = 0 & coefficient of \\(x^2\\) = coefficient of \\(y^2\\).<\/p>\n

Example<\/span><\/strong> : Find the equation of the circle through the points of intersection of \\(x^2 + y^2 – 1\\) = 0, \\(x^2 + y^2 – 2x – 4y + 1\\) = 0 and touching the line x + 2y = 0.<\/p>\n

Solution<\/strong><\/span> : Family of circles is \\(x^2 + y^2 – 2x – 4y + 1\\) + \\(\\lambda\\)(\\(x^2 + y^2 – 1\\)) = 0<\/p>\n

(1 + \\(\\lambda\\))\\(x^2\\) + (1 + \\(\\lambda\\))\\(y^2\\) – 2x – 4y + (1 – \\(\\lambda\\))) = 0<\/p>\n

\\(x^2 + y^2 – {2\\over {1 + \\lambda}} x – {4\\over {1 + \\lambda}}y + {{1 – \\lambda}\\over {1 + \\lambda}}\\) = 0<\/p>\n

Centre is (\\({1\\over {1 + \\lambda}}\\), \\({2\\over {1 + \\lambda}}\\))\u00a0 and radius = \\(\\sqrt{4 + {\\lambda}^2}\\over |1 + \\lambda|\\)<\/p>\n

Since it touches the line x + 2y = 0, hence<\/p>\n

Radius = Perpendicular distance from center to the line<\/p>\n

i.e., |\\({1\\over {1 + \\lambda}} + 2{2\\over {1 + \\lambda}}\\over \\sqrt{1^2 + 2^2}\\)| = \\(\\sqrt{4 + {\\lambda}^2}\\over |1 + \\lambda|\\) \\(\\implies\\) \\(\\sqrt{5}\\) = \\(\\sqrt{4 + {\\lambda}^2}\\) \\(\\implies\\) \\(\\lambda\\) = \\(\\pm\\) 1.<\/p>\n

\\(\\lambda\\) = -1 cannot be possible in case of circle. So \\(\\lambda\\) = 1.<\/p>\n

Hence the equation of the circle is \\(x^2 + y^2 – x – 2y\\) = 0<\/p>\n\n\n

\n
Previous – Common Tangent to Two Circles \u2013 Direct & Transverse<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn equation of family of circles with examples. Let’ begin – Family of Circles (a)\u00a0 The equation of the family of circles passing through the point of intersection of two circles \\(S_1\\) = 0 & \\(S_2\\) = 0 is : \\(S_1\\) + K\\(S_2\\) = 0\u00a0 \u00a0 \u00a0 (K \\(\\ne\\) -1) (b)\u00a0 The …<\/p>\n

Family of Circles – Equations & Examples<\/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":[181,178,179],"yoast_head":"\nFamily of Circles - Equations & Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is family of circles and different equations of family of circles with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/family-of-circles\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Family of Circles - 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