{"id":4820,"date":"2021-08-29T11:58:15","date_gmt":"2021-08-29T11:58:15","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4820"},"modified":"2022-01-16T16:58:00","modified_gmt":"2022-01-16T11:28:00","slug":"distance-of-a-point-from-a-line","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/distance-of-a-point-from-a-line\/","title":{"rendered":"Distance of a Point from a Line – Formula and Example"},"content":{"rendered":"

Here you will learn formula to find the distance of a point from a line with examples.<\/p>\n

Let’s begin –<\/p>\n

Distance of a Point from a Line<\/h2>\n

The length of the perpendicular from a point \\((x_1, y_1)\\)<\/strong> to a line ax + by + c = 0<\/strong> is<\/p>\n

\n

|\\(ax_1 + by_1 + c\\over {\\sqrt{a^2+b^2}}\\)|.<\/p>\n<\/blockquote>\n

It is the distance of a point from a line.<\/p>\n

Distance of a Line from Origin<\/strong><\/h4>\n

The length of the perpendicular from the origin to a line ax + by + c = 0<\/strong> is<\/p>\n

\n

\\( | c |\\over {\\sqrt{a^2+b^2}}\\).<\/p>\n<\/blockquote>\n

Algorithm to find distance :<\/strong><\/p>\n

Step 1 : Write the equation of the line in the form ax + by + c = 0<\/p>\n

Step 2 : Substitute the coordinates \\(x_1\\) and \\(y_1\\) of the point in place of x and y respectively in the expression.<\/p>\n

Step 3 : Divide the result obtained in step 2 by the square root of the sum of the squares of the coefficients of x and y.<\/p>\n

Step 4 : Take the modulus of the expression obtained in step 3.<\/p>\n

he result obtained after step 4 is the required distance.<\/p>\n\n\n

Example : <\/span>Find the distance between the line 12x – 5y + 9 = 0 and the point (2,1).<\/p>\n

Solution : <\/span>We have line 12x – 5y + 9 = 0 and the point (2,1)

\nRequired distance = |\\(12*2 – 5*1 + 9\\over {\\sqrt{12^2 + (-5)^2}}\\)|

\n= \\(|24-5+9|\\over 13\\) = \\(28\\over 13\\)

<\/p>\n\n\n\n

Example : <\/span>If p is the length of the perpendicular from the origin to the line \\(x\\over a\\) + \\(y\\over b\\) = 1, then prove that \\(1\\over p^2\\) = \\(1\\over a^2\\) + \\(1\\over b^2\\)<\/p>\n

Solution : <\/span>The given line is bx + ay – ab = 0 ………….(i)

\nIt is given that

\np = Length of the perpendicular from the origin to line (i)

\n\\(\\implies\\) p = \\(|b(0) + a(0) – ab|\\over {\\sqrt{b^2+a^2}}\\) = \\(ab\\over \\sqrt{a^2+b^2}\\)

\n\\(\\implies\\) \\(p^2\\) = \\(a^2b^2\\over a^2+b^2\\) \\(\\implies\\) \\(1\\over p^2\\) = \\(a^2+b^2\\over a^2b^2\\) \\(\\implies\\) \\(1\\over p^2\\) = \\(1\\over a^2\\) + \\(1\\over b^2\\)

\nHence Proved.

<\/p>\n\n\n\n

\n
Previous – Point of Intersection of Two Lines <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn formula to find the distance of a point from a line with examples. Let’s begin – Distance of a Point from a Line The length of the perpendicular from a point \\((x_1, y_1)\\) to a line ax + by + c = 0 is |\\(ax_1 + by_1 + c\\over {\\sqrt{a^2+b^2}}\\)|. It …<\/p>\n

Distance of a Point from a Line – Formula and Example<\/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":[32],"tags":[634,633],"yoast_head":"\nDistance of a Point from a Line - Formula and Example - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn formula to find the distance of a point from a line and distance of a line from origin 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\/distance-of-a-point-from-a-line\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Distance of a Point from a Line - 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