{"id":8335,"date":"2021-11-20T16:22:04","date_gmt":"2021-11-20T10:52:04","guid":{"rendered":"https:\/\/mathemerize.com\/?p=8335"},"modified":"2022-02-01T00:45:34","modified_gmt":"2022-01-31T19:15:34","slug":"what-is-the-director-circle-of-a-circle","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-director-circle-of-a-circle\/","title":{"rendered":"Director Circle of a Circle – Equation and Proof"},"content":{"rendered":"

Here you will learn what is the equation of director circle of circle with proof.<\/p>\n

Let’s begin –<\/p>\n

Director Circle of a Circle<\/h2>\n

The equation of director circle is \\(x^2\\) + \\(y^2\\) = 2\\(a^2\\).<\/strong><\/p><\/blockquote>\n

Proof<\/strong> : The locus of the point of intersection of two perpendicular tangents to the circle is called director circle. Let P(h, k) is the point of intersection of two tangents drawn on the circle (\\(x^2\\) + \\(y^2\\) = \\(a^2\\)). Then the equation of the pair of tangents<\/a> is \\(SS_1 = T^2\\).<\/p>\n

i.e. (\\(x^2\\) + \\(y^2\\) – \\(a^2\\))(\\(h^2\\) + \\(k^2\\) – \\(a^2\\)) = \\({(hx + ky – a^2)}^2\\)<\/p>\n

As lines are perpendicular to each other then, coefficient of \\(x^2\\) + coefficient of \\(y^2\\) = 0.<\/p>\n

\\(\\implies\\)   [(\\(h^2\\) + \\(k^2\\) – \\(a^2)-h^2\\)][(\\(h^2\\) + \\(k^2\\) – \\(a^2)-k^2\\)] = 0<\/p>\n

\\(\\implies\\)  \\(h^2\\) + \\(k^2\\) = 2\\(a^2\\)<\/p>\n

\\(\\therefore\\)  locus of (h, k) is \\(x^2\\) + \\(y^2\\) = 2\\(a^2\\)<\/strong> which is the equation of the director circle<\/strong><\/p>\n

\\(\\therefore\\)  director circle is a concentric circle whose radius is \\(\\sqrt{2}\\) times the radius of the circle.<\/p>\n

Note<\/strong> : The director circle of \\(x^2 + y^2 + 2gx + 2fy + c = 0\\) is<\/p>\n

\\(x^2 + y^2 + 2gx + 2fy + 2c – g^2 – f^2\\) = 0<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

Here you will learn what is the equation of director circle of circle with proof. Let’s begin – Director Circle of a Circle The equation of director circle is \\(x^2\\) + \\(y^2\\) = 2\\(a^2\\). Proof : The locus of the point of intersection of two perpendicular tangents to the circle is called director circle. Let …<\/p>\n

Director Circle of a Circle – Equation and Proof<\/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":[858,857],"yoast_head":"\nDirector Circle of a Circle - Equation and Proof<\/title>\n<meta name=\"description\" content=\"In this post you will learn what is the equation of director circle of circle with proof.\" \/>\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\/what-is-the-director-circle-of-a-circle\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Director Circle of a Circle - 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