{"id":10069,"date":"2022-02-15T18:06:38","date_gmt":"2022-02-15T12:36:38","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10069"},"modified":"2022-02-17T18:41:07","modified_gmt":"2022-02-17T13:11:07","slug":"foci-of-hyperbola-formula-and-coordinates","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/foci-of-hyperbola-formula-and-coordinates\/","title":{"rendered":"Foci of Hyperbola Formula and Coordinates"},"content":{"rendered":"

Here you will learn how to find the coordinates of the foci of hyperbola with examples.<\/p>\n

Let’s begin –<\/p>\n

Foci of Hyperbola Coordinates<\/h2>\n

(i) For the hyperbola \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1<\/strong><\/h4>\n

The coordinates of foci are (ae, 0) and (-ae, 0).<\/p><\/blockquote>\n

(ii) For the conjugate hyperbola -\\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1<\/strong><\/h4>\n

The coordinates of foci are (0, be) and (0, -be).<\/p><\/blockquote>\n

Also Read<\/strong> : Equation of the Hyperbola | Graph of a Hyperbola<\/a><\/p>\n

Example<\/strong><\/span> : For the given hyperbola, find the coordinates of foci<\/p>\n

(i)  \\(16x^2 – 9y^2\\) = 144<\/p>\n

(ii)  \\(9x^2 – 16y^2 – 18x + 32y – 151\\) = 0<\/p>\n

Solution<\/strong><\/span> :<\/p>\n

(i)<\/strong>  We have,<\/p>\n

\\(16x^2 – 9y^2\\) = 144 \\(\\implies\\) \\(x^2\\over 9\\) – \\(y^2\\over 16\\) = 1,<\/p>\n

This is of the form \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1<\/p>\n

where \\(a^2\\) = 9 and \\(b^2\\) = 16 i.e. a = 3 and b = 4<\/p>\n

Eccentricity (e) = \\(\\sqrt{1 + {b^2\\over a^2}}\\)<\/p>\n

\\(\\implies\\) e = \\(\\sqrt{1 + 16\/9}\\) = \\(5\\over 3\\)<\/p>\n

Therefore, the coordinates of foci are (ae, 0) and (-ae, 0).<\/p>\n

\\(\\implies\\)  (5, 0) and (-5, 0).<\/p>\n

(ii) <\/strong>We have,<\/p>\n

\\(9x^2 – 16y^2 – 18x + 32y – 151\\) = 0<\/p>\n

\\(\\implies\\) \\(9(x^2 – 2x)\\) – \\(16(y^2 – 2y)\\) = 151<\/p>\n

\\(\\implies\\)  \\(9(x^2 – 2x + 1)\\) – \\(16(y^2 – 2y + 1)\\) = 144<\/p>\n

\\(\\implies\\)  \\(9(x – 1)^2\\) – \\(16(y – 1)^2\\) = 144<\/p>\n

\\(\\implies\\)  \\((x – 1)^2\\over 16\\) – \\((y – 1)^2\\over 9\\) = 1<\/p>\n

Here, a = 4 and b = 3<\/p>\n

Eccentricity (e) = \\(\\sqrt{1 + {b^2\\over a^2}}\\)<\/p>\n

\\(\\implies\\) e = \\(\\sqrt{1 + 9\/16}\\) = \\(5\\over 4\\)<\/p>\n

Here, the center is (h, k) i.e. (1, 1)<\/p>\n

Therefore, the coordinates of foci are (ae + h, k) and (-ae + h, k).<\/p>\n

\\(\\implies\\)  (6, 1) and (-4, 1).<\/p>\n

Note<\/strong> : For the hyperbola \\((x – h)^2\\over a^2\\) – \\((y – k)^2\\over b^2\\) = 1 with center (h. k),<\/p>\n

(i)<\/strong> For normal hyperbola,<\/p>\n

The coordinates of foci are (ae + h, k) and (-ae + h, k).<\/p><\/blockquote>\n

(ii)<\/strong> For conjugate hyperbola,<\/p>\n

The coordinates of foci are (h, be + k) and (h, -be + k).<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find the coordinates of the foci of hyperbola with examples. Let’s begin – Foci of Hyperbola Coordinates (i) For the hyperbola \\(x^2\\over a^2\\) – \\(y^2\\over b^2\\) = 1 The coordinates of foci are (ae, 0) and (-ae, 0). (ii) For the conjugate hyperbola -\\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = …<\/p>\n

Foci of Hyperbola Formula and Coordinates<\/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":[25],"tags":[],"yoast_head":"\nFoci of Hyperbola Formula and Coordinates - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find the coordinates of the foci of hyperbola 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\/foci-of-hyperbola-formula-and-coordinates\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Foci of Hyperbola Formula and Coordinates - 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