{"id":5986,"date":"2021-10-04T21:11:40","date_gmt":"2021-10-04T15:41:40","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5986"},"modified":"2021-10-04T21:12:01","modified_gmt":"2021-10-04T15:42:01","slug":"ellipse-examples","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/ellipse-examples\/","title":{"rendered":"Ellipse Examples"},"content":{"rendered":"

Here you will learn some ellipse examples for better understanding of ellipse concepts<\/a><\/span>.<\/p>\n\n\n

\n
\n\n \n

Example 1 : <\/span> Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \\(\\sqrt{5}\\)<\/p>\n

Solution : <\/span>Here S = (2, 3) & S’ is (-2, 3) and b = \\(\\sqrt{5}\\) \\(\\implies\\) SS’ = 4 = 2ae \\(\\implies\\) ae = 2

\n but \\(b^2\\) = \\(a^2(1-e^2)\\) \\(\\implies\\) 5 = \\(a^2\\) – 4 \\(\\implies\\) a = 3

\n Hence the equation to major axis is y = 3.

\n Centre of ellipse is midpoint of SS’ i.e. (0, 3)

\n \\(\\therefore\\)   Equation to ellipse is \\(x^2\\over a^2\\) + \\({(y-3)}^2\\over b^2\\) = 1 or \\(x^2\\over 9\\) + \\({(y-3)}^2\\over 5\\) = 1<\/p>\n


\n\n \n

Example 2 : <\/span> For what value of k does the line y = x + k touches the ellipse \\(9x^2 + 16y^2\\) = 144.<\/p>\n

Solution : <\/span>\\(\\because\\) Equation of ellipse is \\(9x^2 + 16y^2\\) = 144 or \\(x^2\\over 16\\) + \\({(y-3)}^2\\over 9\\) = 1

\n comparing this with \\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1 then we get \\(a^2\\) = 16 and \\(b^2\\) = 9

\n and comparing the line y = x + k with y = mx + c   m = 1 and c = k

\n If the line y = x + k touches the ellipse \\(9x^2 + 16y^2\\) = 144, then \\(c^2\\) = \\(a^2m^2 + b^2\\)

\n \\(\\implies\\) \\(k^2\\) = 16 \\(\\times\\) \\(1^2\\) + 9 \\(\\implies\\) \\(k^2\\) = 25

\n   \\(\\therefore\\)   k = \\(\\pm\\)5\n <\/p>


\n\n \n

Example 3 : <\/span> Find the equation of the tangents to the ellipse \\(3x^2+4y^2\\) = 12 which are perpendicular to the line y + 2x = 4<\/p>\n

Solution : <\/span>Let m be the slope of the tangent, since the tangent is perpendicular to the line y + 2x = 4

\\(\\therefore\\)   mx – 2 = -1 \\(\\implies\\) m = \\(1\\over 2\\)

Since \\(3x^2+4y^2\\) = 12 or \\(x^2\\over 4\\) + \\(y^2\\over 3\\) = 1

Comparing this with \\(x^2\\over a^2\\) + \\(y^2\\over b^2\\) = 1

\\(\\therefore\\)   \\(a^2\\) = 4 and \\(b^2\\) = 3

So the equation of the tangent are y = \\(1\\over 2\\)x \\(\\pm\\) \\(\\sqrt{4\\times {1\\over 4} + 3}\\)

\\(\\implies\\) y = \\(1\\over 2\\)x \\(\\pm\\) 2 or x – 2y \\(\\pm\\) 4 = 0<\/p>\n
\n

Practice these given ellipse examples to test your knowledge on concepts of ellipse.<\/p>\n \n <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn some ellipse examples for better understanding of ellipse concepts. Example 1 : Find the equation of ellipse whose foci are (2, 3), (-2, 3) and whose semi major axis is of length \\(\\sqrt{5}\\) Solution : Here S = (2, 3) & S’ is (-2, 3) and b = \\(\\sqrt{5}\\) \\(\\implies\\) SS’ …<\/p>\n

Ellipse 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":[24],"tags":[],"yoast_head":"\nEllipse Examples - Mathemerize<\/title>\n<meta name=\"description\" content=\"Solve these Ellipse examples to practice and test your knowledge of Ellipse concepts.\" \/>\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\/ellipse-examples\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Ellipse Examples - 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