{"id":8327,"date":"2021-11-20T16:15:32","date_gmt":"2021-11-20T10:45:32","guid":{"rendered":"https:\/\/mathemerize.com\/?p=8327"},"modified":"2022-02-01T00:32:54","modified_gmt":"2022-01-31T19:02:54","slug":"what-are-orthogonal-circles-and-condition-of-orthogonal-circles","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-are-orthogonal-circles-and-condition-of-orthogonal-circles\/","title":{"rendered":"Orthogonal Circles and Condition of Orthogonal Circles"},"content":{"rendered":"

Here you will learn what are orthogonal circles and condition of orthogonal circles.<\/p>\n

Let’s begin –<\/p>\n

Orthogonal Circles<\/h2>\n

Let two circles are \\(S_1\\) = \\({x}^2 + {y}^2 + 2{g_1}x + 2{f_1}y + {c_1}\\) = 0 and \\(S_2\\) = \\({x}^2 + {y}^2 + 2{g_2}x + 2{f_2}y + {c_2}\\) = 0.Then<\/p>\n

Angle of intersection of two circles is cos\\(\\theta\\) = |\\(2{g_1}{g_2} + 2{f_1}{f_2} – {c_1} – {c_2}\\over {2\\sqrt{{g_1}^2 + {f_1}^2 -c_1}}{\\sqrt{{g_1}^2 + {f_1}^2 -c_1}}\\)|<\/p>\n

If the angle of intersection of the two circles<\/a> is a right angle<\/strong> then such circles are called ‘Orthogonal circles’\u00a0<\/strong><\/p>\n

i.e. \\(\\theta\\) = 90 \\(\\implies\\)\u00a0 \u00a0\\(cos\\theta\\) = 0<\/p>\n

\\(\\implies\\)\u00a0 |\\(2{g_1}{g_2} + 2{f_1}{f_2} – {c_1} – {c_2}\\over {2\\sqrt{{g_1}^2 + {f_1}^2 -c_1}}{\\sqrt{{g_1}^2 + {f_1}^2 -c_1}}\\)| = 0<\/p>\n

Orthogonal Circles Condition<\/strong><\/h4>\n

\\(2g_1g_2 + 2f_1f_2 = c_1 + c_2\\)<\/strong><\/p><\/blockquote>\n

Example<\/strong><\/span> : Show that the circles \\(x^2 + y^2 + 4x + 6y + 3\\) = 0 and \\(2x^2 + 2y^2 + 6x + 4y + 18 = 0\\) intersect orthogonally.<\/p>\n

Solution<\/strong><\/span> : We have given two circles,<\/p>\n

\\(x^2 + y^2 + 4x + 6y + 3\\) = 0 and \\(2x^2 + 2y^2 + 6x + 4y + 18 = 0\\)<\/p>\n

Write second circle in standard form by dividing by 2,<\/p>\n

\\(x^2 + y^2 + 3x + 2y + 9\\) = 0<\/p>\n

Now, comparing these two equations with above equations \\(S_1\\) and \\(S_2\\),<\/p>\n

From first equation,<\/p>\n

\\(g_1\\) = 2, \\(f_1\\) = 3 and \\(c_1\\) = 3<\/p>\n

From second equation,<\/p>\n

\\(g_2\\) = \\(3\\over 2\\), \\(f_2\\) = 1 and \\(c_2\\) = 9<\/p>\n

Condition to prove orthogonal,<\/p>\n

\\(2g_1g_2 + 2f_1f_2 = c_1 + c_2\\)<\/p>\n

2(2)(\\(3\\over 2\\)) + 2(3)(1) = 3 + 9<\/p>\n

6 + 6 = 3 + 9<\/p>\n

12 = 12<\/p>\n

L.H.S = R.H.S<\/p>\n

Hence, both the circle intersect orthogonally.<\/p>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what are orthogonal circles and condition of orthogonal circles. Let’s begin – Orthogonal Circles Let two circles are \\(S_1\\) = \\({x}^2 + {y}^2 + 2{g_1}x + 2{f_1}y + {c_1}\\) = 0 and \\(S_2\\) = \\({x}^2 + {y}^2 + 2{g_2}x + 2{f_2}y + {c_2}\\) = 0.Then Angle of intersection of two circles …<\/p>\n

Orthogonal Circles and Condition of Orthogonal Circles<\/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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[23],"tags":[855,856],"yoast_head":"\nOrthogonal Circles and Condition of Orthogonal Circles<\/title>\n<meta name=\"description\" content=\"In this post you will learn what are orthogonal circles and condition of orthogonal circles.\" \/>\n<meta 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