{"id":8071,"date":"2021-11-13T20:01:42","date_gmt":"2021-11-13T14:31:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=8071"},"modified":"2021-11-14T00:58:23","modified_gmt":"2021-11-13T19:28:23","slug":"find-the-angle-between-the-curves-xy-6-and-x2-y-12","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/find-the-angle-between-the-curves-xy-6-and-x2-y-12\/","title":{"rendered":"Find the angle between the curves xy = 6 and \\(x^2 y\\) =12."},"content":{"rendered":"

Solution :<\/h2>\n

The equation of the two curves are<\/p>\n

xy = 6\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …….(i)<\/p>\n

and, \\(x^2 y\\) = 12\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …………(ii)<\/p>\n

from (i) , we obtain y = \\(6\\over x\\). Putting this value of y in (ii), we obtain<\/p>\n

\\(x^2\\) \\((6\\over x)\\) = 12 \\(\\implies\\) 6x = 12<\/p>\n

\\(\\implies\\) x = 2<\/p>\n

Putting x = 2 in (i)\u00a0 or (ii), we get y = 3.<\/p>\n

Thus, the two curves intersect at P(2, 3).<\/p>\n

Differentiating (i) with respect to x, we get<\/p>\n

x\\(dy\\over dx\\) + y = 0 \\(\\implies\\) \\(dy\\over dx\\) = \\(-y\\over x\\)<\/p>\n

\\(\\implies\\) \\(m_1\\) = \\(({dy\\over dx})_{(2, 3)}\\) = \\(-3\\over 2\\)<\/p>\n

Differentiating (ii) with respect to x, we get<\/p>\n

\\(x^2\\) \\(dy\\over dx\\) + 2xy\u00a0 = 0 \\(\\implies\\) \\(dy\\over dx\\) = \\(-2y\\over x\\)<\/p>\n

\\(\\implies\\) \\(m_2\\) = \\(({dy\\over dx})_{(2, 3)}\\) = -3<\/p>\n

Let \\(\\theta\\) be the angle, then angle between angle between two curves<\/a><\/p>\n

\\(tan \\theta\\) = \\(m_1 – m_2\\over 1 + m_1 m_2\\) = \\(3\\over 11\\)<\/p>\n

\\(\\theta\\) = \\(tan^{-1} (3\/11)\\)<\/p>\n

\n

Similar Questions<\/h3>\n

Find the equations of the tangent and the normal at the point \u2018t\u2019 on the curve x = \\(a sin^3 t\\), y = \\(b cos^3 t\\).<\/a><\/p>\n

Find the equation of the normal to the curve y = \\(2x^2 + 3 sin x\\) at x = 0.<\/a><\/p>\n

Find the equation of the tangent to curve y = \\(-5x^2 + 6x + 7\\)\u00a0 at the point (1\/2, 35\/4).<\/a><\/p>\n

Check the orthogonality of the curves \\(y^2\\) = x and \\(x^2\\) = y.<\/a><\/p>\n

The angle of intersection between the curve \\(x^2\\) = 32y and \\(y^2\\) = 4x at point (16, 8) is<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : The equation of the two curves are xy = 6\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …….(i) and, \\(x^2 y\\) = 12\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …………(ii) from (i) , we obtain y = \\(6\\over x\\). Putting this value of y in (ii), we obtain \\(x^2\\) \\((6\\over x)\\) = 12 \\(\\implies\\) 6x = 12 …<\/p>\n

Find the angle between the curves xy = 6 and \\(x^2 y\\) =12.<\/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":[66,43],"tags":[],"yoast_head":"\nFind the angle between the curves xy = 6 and \\(x^2 y\\) =12.<\/title>\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\/find-the-angle-between-the-curves-xy-6-and-x2-y-12\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Find the angle between the curves xy = 6 and \\(x^2 y\\) =12.\" \/>\n<meta property=\"og:description\" content=\"Solution : The equation of the two curves are xy = 6\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …….(i) and, (x^2 y) = 12\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 …………(ii) from (i) , we obtain y = (6over x). Putting this value of y in (ii), we obtain (x^2) ((6over x)) = 12 (implies) 6x = 12 … Find the angle between the curves xy = 6 and (x^2 y) =12. 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