{"id":5593,"date":"2021-09-22T16:09:05","date_gmt":"2021-09-22T10:39:05","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5593"},"modified":"2021-11-13T21:35:53","modified_gmt":"2021-11-13T16:05:53","slug":"equation-of-tangent-and-normal-to-the-curve","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/equation-of-tangent-and-normal-to-the-curve\/","title":{"rendered":"Equation of Tangent and Normal to the Curve"},"content":{"rendered":"

Here you will learn equation of tangent and normal to the curve with examples.<\/p>\n

Let’s begin –<\/p>\n

Equation of Tangent and Normal to the Curve<\/h2>\n

We know that the equation of line passing through a point \\((x_1, y_1)\\)<\/strong> and having slope m is \\(y – y_1\\) = m\\((x – x_1)\\).<\/strong><\/p>\n

and we know that the slopes of the tangent and the normal to the curve y = f(x) at a point P\\((x_1, y_1)\\) are \\(({dy\\over dx})_P\\)<\/strong> and -\\(1\\over ({dy\\over dx})_P\\)<\/strong> respectively. <\/p>\n

Therefore the equation of the tangent <\/strong>at P\\((x_1, y_1)\\) to the curve y = f(x) is<\/p>\n

\n

\\(y – y_1\\) = \\(({dy\\over dx})_P\\) (\\(x – x_1\\))<\/p>\n<\/blockquote>\n

Since the normal at P\\((x_1, y_1)\\) passes through P and has slope -\\(1\\over ({dy\\over dx})_P\\).<\/p>\n

Therefore, the equation of the normal<\/strong> at P\\((x_1, y_1)\\) to the curve y = f(x) is<\/p>\n

\n

\\(y – y_1\\) = \\(-1\\over ({dy\\over dx})_P\\) (\\(x – x_1\\))<\/p>\n<\/blockquote>\n

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

1). If \\(({dy\\over dx})_P\\) = \\(\\pm \\infty\\), then the tangent at \\((x_1, y_1)\\)  is parallel to y-axis and its equation is x = \\(x_1\\).<\/p>\n

2). If \\(({dy\\over dx})_P\\) = 0, then the normal at \\((x_1, y_1)\\)  is parallel to y-axis and its equation is x = \\(x_1\\).<\/p>\n

3). If \\(({dy\\over dx})_P\\) = \\(\\pm \\infty\\), then the normal at \\((x_1, y_1)\\)  is parallel to x-axis and its equation is y = \\(y_1\\).<\/p>\n

4). If \\(({dy\\over dx})_P\\) = 0, then the tangent at \\((x_1, y_1)\\)  is parallel to x-axis and its equation is y = \\(y_1\\).<\/p>\n

Example<\/span><\/strong> : find the equation of the tangent to curve y = \\(-5x^2 + 6x + 7\\)  at the point (1\/2, 35\/4).<\/p>\n

Solution<\/span><\/strong> : The equation of the given curve is <\/p>\n

y = \\(-5x^2 + 6x + 7\\) <\/p>\n

\\(\\implies\\) \\(dy\\over dx\\) = -10x + 6<\/p>\n

\\(\\implies\\) \\(({dy\\over dx})_{(1\/2, 35\/4)}\\) = \\(-10\\over 4\\) + 6 = 1<\/p>\n

The required equation at (1\/2, 35\/4) is<\/p>\n

y – \\(35\\over 4\\) = \\(({dy\\over dx})_{(1\/2, 35\/4)}\\) \\((x – {1\\over 2})\\)<\/p>\n

\\(\\implies\\) y – 35\/4 = 1(x – 1\/2)<\/p>\n

\\(\\implies\\) y = x + 33\/4<\/p>\n

\n

Related 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\n\n

\n
Next – Angle of Intersection of Two Curves<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Slopes of Tangent and Normal to the Curve<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn equation of tangent and normal to the curve with examples. Let’s begin – Equation of Tangent and Normal to the Curve We know that the equation of line passing through a point \\((x_1, y_1)\\) and having slope m is \\(y – y_1\\) = m\\((x – x_1)\\). and we know that the …<\/p>\n

Equation of Tangent and Normal to the Curve<\/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":[37],"tags":[76,98,95,97,96],"yoast_head":"\nEquation of Tangent and Normal to the Curve - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn equation of tangent and normal to the curve 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\/equation-of-tangent-and-normal-to-the-curve\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equation of Tangent and Normal to the Curve - 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