{"id":3505,"date":"2021-07-28T12:29:11","date_gmt":"2021-07-28T12:29:11","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3505"},"modified":"2021-11-24T23:20:24","modified_gmt":"2021-11-24T17:50:24","slug":"parametric-equation-of-rectangular-hyperbola","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/parametric-equation-of-rectangular-hyperbola\/","title":{"rendered":"Parametric Equation of Rectangular Hyperbola"},"content":{"rendered":"

Here, you will learn what is rectangular hyperbola and parametric equation of rectangular hyperbola with example.<\/p>\n

Let’s begin –<\/p>\n

Rectangular Hyperbola<\/h2>\n

The particular kind of the hyperbola in which the length of it’s transverse axis and conjugate axis are equal is called rectangular hyperbola. The eccentricity of the rectangular hyperbola<\/strong> is \\(\\sqrt{2}\\) and the length of it’s latus rectum is equal to it’s transverse axis or conjugate axis.<\/p>\n

\n

The equation of rectangular hyperbola is xy = \\(c^2\\).<\/p>\n<\/blockquote>\n

Parametric Equation of Rectangular Hyperbola<\/h2>\n

The Rectangular hyperbola referred to its asymptotes as axis of coordinates.<\/p>\n

\n

The parametric equation of the rectangular hyperbola xy = \\(c^2\\) with parametric representation is x = ct, y = c\/t, t \\(\\in\\) R – {0}.<\/p>\n<\/blockquote>\n

Basic Definitions of Rectangular Hyperbola :<\/h2>\n

For the hyperbola, xy = \\(c^2\\)<\/p>\n

\n

(i) Vertices : (c, c) & (-c, -c).<\/p>\n

(ii) Foci : (\\(\\sqrt{2c}, \\sqrt{2c}\\)) & (\\(-\\sqrt{2c}, -\\sqrt{2c}\\))<\/p>\n

(iii) Directrices : x + y = \\(\\pm \\sqrt{2c}\\)<\/p>\n

(iv) Latus rectum : l = \\(2\\sqrt{2c}\\) = T.A = C.A<\/p>\n<\/blockquote>\n

Note :<\/strong><\/p>\n

(a) The equation of chord joining the points \\(P(t_1)\\) & \\(Q(t_2)\\) is x + \\(t_1t_2\\)y = c(\\(t_1+t_2\\)) with slope,
m = \\(-1\\over t_1t_2\\).<\/p>\n

(b) The equation of the tangent to rectangular hyperbola in point form at P(\\(x_1, y_1\\)) is \\(x\\over x_1\\) + \\(y\\over y_1\\) = 2 & in parametric form at P(t) is \\(x\\over t\\) + ty = 2c.<\/p>\n

(c) Equation of normal in parametric form is y – \\(c\\over t\\) = \\(t^2\\)(x – ct).<\/p>\n

(d) The equation of chord with a given middle point as (h, k) is kx + hy = 2hk.<\/p>\n\n\n

Example : <\/span> Find the parametric representation and equation of tangent at the point (1, 2) to the rectangular hyperbola xy = 2.<\/p>\n

Solution : <\/span>Since, the equation of rectangular hyperbola is xy = 2.

\nComparing with the equation of rectangular hyperbola xy = \\(c^2\\).

\nwe get c = \\(\\sqrt{2}\\)

\nEquation in parametric representation x = ct and y = c\/t.

\n\\(\\implies\\) x = \\(\\sqrt{2}\\)t and y = \\(\\sqrt{2}\\)\/t

\nEquation of the tangent to rectangular hyperbola in point form at P(\\(x_1, y_1\\)) is \\(x\\over x_1\\) + \\(y\\over y_1\\) = 2

\n\\(\\implies\\) \\(x\\over 1\\) + \\(y\\over 2\\) = 2

<\/p>\n\n\n

Hope you learnt what is the equation of rectangular hyperbola, learn more concepts of hyperbola and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – Equation of Tangent to Hyperbola in all Forms<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – What is the Equation of Conjugate Hyperbola <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn what is rectangular hyperbola and parametric equation of rectangular hyperbola with example. Let’s begin – Rectangular Hyperbola The particular kind of the hyperbola in which the length of it’s transverse axis and conjugate axis are equal is called rectangular hyperbola. The eccentricity of the rectangular hyperbola is \\(\\sqrt{2}\\) and the length …<\/p>\n

Parametric Equation of Rectangular Hyperbola<\/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":[25],"tags":[388,385,386,387],"yoast_head":"\nParametric Equation of Rectangular Hyperbola - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn what is the parametric equation of rectangular hyperbola and its basic concepts 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\/parametric-equation-of-rectangular-hyperbola\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Parametric Equation of Rectangular Hyperbola - 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