{"id":10058,"date":"2022-02-15T17:50:50","date_gmt":"2022-02-15T12:20:50","guid":{"rendered":"https:\/\/mathemerize.com\/?p=10058"},"modified":"2022-02-17T17:34:58","modified_gmt":"2022-02-17T12:04:58","slug":"foci-of-ellipse-formula-and-coordinates","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/foci-of-ellipse-formula-and-coordinates\/","title":{"rendered":"Foci of Ellipse Formula and Coordinates"},"content":{"rendered":"

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

Let’s begin –<\/p>\n

Foci of Ellipse Formula and Coordinates<\/h2>\n

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

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

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

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

Also Read<\/strong> : Different Types of Ellipse Equations and Graph<\/a><\/p>\n

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

(i)\u00a0 \\(16x^2 + 25y^2\\) = 400<\/p>\n

(ii)\u00a0 \\(x^2 + 4y^2 – 2x\\) = 0<\/p>\n

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

(i)<\/strong>\u00a0 We have,<\/p>\n

\\(16x^2 + 25y^2\\) = 400 \\(\\implies\\) \\(x^2\\over 25\\) + \\(y^2\\over 16\\),<\/p>\n

where \\(a^2\\) = 25 and \\(b^2\\) = 16 i.e. a = 5 and b = 4<\/p>\n

Clearly a > b,<\/p>\n

The eccentricity of ellipse (e) = \\(\\sqrt{1 – {b^2\\over a^2}}\\)<\/p>\n

e = \\(\\sqrt{1 – 16\/25}\\) = \\(3\\over 5\\)<\/p>\n

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

\\(\\implies\\) (3, 0) and (-3, 0).<\/p>\n

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

\\(x^2 + 4y^2 – 2x\\) = 0<\/p>\n

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

\\(\\implies\\)\u00a0 \\((x – 1)^2\\over 1^2\\) + \\((y – 0)^2\\over (1\/2)^2\\) = 1<\/p>\n

Here, a = 1 and b = 1\/2<\/p>\n

Clearly a > b,<\/p>\n

The eccentricity of ellipse (e) = \\(\\sqrt{1 – {b^2\\over a^2}}\\)<\/p>\n

e = \\(\\sqrt{1 – 1\/4}\\) = \\(\\sqrt{3}\\over 2\\)<\/p>\n

Since, Center of the above ellipse (h, k) i.e. (1, 0)<\/p>\n

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

\\(\\implies\\) (\\(\\sqrt{3}\\over 2\\) + 1, 0) and (-\\(\\sqrt{3}\\over 2\\) + 1, 0).<\/p>\n

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

(i)<\/strong> For ellipse a > b,<\/p>\n

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

(ii)<\/strong> For ellipse a < b,<\/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 ellipse formula with examples. Let’s begin – Foci of Ellipse Formula and Coordinates (i) For the ellipse \\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1, a > b The coordinates of foci are (ae, 0) and (-ae, 0) (ii) For the ellipse \\(x^2\\over …<\/p>\n

Foci of Ellipse 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":[24],"tags":[],"yoast_head":"\nFoci of Ellipse 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 ellipse formula 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-ellipse-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 Ellipse Formula and Coordinates - 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