{"id":6761,"date":"2021-10-21T13:56:11","date_gmt":"2021-10-21T08:26:11","guid":{"rendered":"https:\/\/mathemerize.com\/?p=6761"},"modified":"2021-10-25T01:16:15","modified_gmt":"2021-10-24T19:46:15","slug":"what-is-the-equation-of-common-tangent-to-the-parabola-y2-4ax-and-x2-4ay","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/what-is-the-equation-of-common-tangent-to-the-parabola-y2-4ax-and-x2-4ay\/","title":{"rendered":"What is the equation of common tangent to the parabola \\(y^2\\) = 4ax and \\(x^2\\) = 4ay ?"},"content":{"rendered":"

Solution :<\/h2>\n

The equation of tangent in slope form to \\(y^2\\) = 4ax is<\/p>\n

y = mx + \\(a\\over m\\)<\/p>\n

Now, if it is common to both parabola, it also lies on second parabola<\/p>\n

then \\(x^2\\) = 4a(mx + \\(a\\over m\\))<\/p>\n

\\(mx^2 – 4am^2 – 4a^2\\) = 0 has equal roots.<\/p>\n

then its discriminant is zero. i.e. \\(b^2-4ac\\) = 0<\/p>\n

\\(16a^2m^2 + 16a^2m\\) = 0<\/p>\n

m = -1<\/p>\n

Putting m = -1 in equation y = mx + \\(a\\over m\\) we get<\/p>\n

y = -x – a<\/p>\n

x + y + a = 0<\/p>\n

Which is the required equation of common tangent to both parabola.<\/p>\n

\n

Similar Questions<\/h3>\n

The slope of the line touching both the parabolas \\(y^2\\) = 4x and \\(x^2\\) = -32 is<\/a><\/p>\n

Find the locus of middle point of the chord of the parabola \\(y^2\\) = 4ax which pass through a given (p, q).<\/a><\/p>\n

Find the equation of the tangents to the parabola \\(y^2\\) = 9x which go through the point (4,10).<\/a><\/p>\n

Find the value of k for which the point (k-1, k) lies inside the parabola \\(y^2\\) = 4x.<\/a><\/p>\n

The length of latus rectum of a parabola, whose focus is (2, 3) and directrix is the line x \u2013 4y + 3 = 0 is<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Solution : The equation of tangent in slope form to \\(y^2\\) = 4ax is y = mx + \\(a\\over m\\) Now, if it is common to both parabola, it also lies on second parabola then \\(x^2\\) = 4a(mx + \\(a\\over m\\)) \\(mx^2 – 4am^2 – 4a^2\\) = 0 has equal roots. then its discriminant is …<\/p>\n

What is the equation of common tangent to the parabola \\(y^2\\) = 4ax and \\(x^2\\) = 4ay ?<\/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":[43,46],"tags":[],"yoast_head":"\nWhat is the equation of common tangent to the parabola \\(y^2\\) = 4ax and \\(x^2\\) = 4ay ?<\/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\/what-is-the-equation-of-common-tangent-to-the-parabola-y2-4ax-and-x2-4ay\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"What is the equation of common tangent to the parabola \\(y^2\\) = 4ax and \\(x^2\\) = 4ay ?\" \/>\n<meta property=\"og:description\" content=\"Solution : The equation of tangent in slope form to (y^2) = 4ax is y = mx + (aover m) Now, if it is common to both parabola, it also lies on second parabola then (x^2) = 4a(mx + (aover m)) (mx^2 – 4am^2 – 4a^2) = 0 has equal roots. then its discriminant is … What is the equation of common tangent to the parabola (y^2) = 4ax and (x^2) = 4ay ? 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