{"id":2771,"date":"2021-07-15T16:17:07","date_gmt":"2021-07-15T16:17:07","guid":{"rendered":"https:\/\/mathemerize.com\/?p=2771"},"modified":"2021-11-30T16:49:31","modified_gmt":"2021-11-30T11:19:31","slug":"concurrency-of-lines","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/concurrency-of-lines\/","title":{"rendered":"Condition of Concurrency of Lines"},"content":{"rendered":"

There are two conditions of concurrency of lines which are given below :<\/p>\n

(a) Three lines are said to be concurrent if they pass through a common point i.e. they meet at a point.<\/p>\n

Thus, Three lines \\(a_1x + b_1y + c_1\\) = 0 and \\(a_2x + b_2y + c_2\\) = 0 and \\(a_3x + b_3y + c_3\\) = 0 are concurrent, if<\/p>\n

\\(\\begin{vmatrix}
a_1 & b_1 & c_1 \\\\
a_2 & b_2 & c_2 \\\\
a_3 & b_3 & c_3 \\\\
\\end{vmatrix}\\) = 0<\/p>\n

This is the required condition of concurrency of lines.<\/p>\n\n\n

Example : <\/span> Prove that the lines 3x + y – 14 = 0, x – 2y = 0 and 3x – 8y + 4 = 0.<\/p>\n

Solution : <\/span>Given lines are 3x + y – 14 = 0, x – 2y = 0 and 3x – 8y + 4 = 0

\nWe have, \\(\\begin{vmatrix}\n3 & 1 & -14 \\\\\n1 & -2 & 0 \\\\\n3 & -8 & 4 \\\\\n\\end{vmatrix}\\) = 3(-8 + 0) – 1(4 – 0) – 14(-8 + 6)

\n= -24 – 4 + 28 = 0

\nSo, the given lines are concurrent.<\/p>\n\n\n

Another Condition of Concurrency of Lines :<\/h2>\n

(b) To test the concurrency of lines, first find out the point of intersection of the three lines. If this point lies on third line ( coordinates of the point satisfy the equation of third line) then the three lines are concurrent otherwise not concurrent.<\/p>\n\n\n

Example : <\/span> Show that the lines x – y – 6 = 0, 4x – 3y – 20 = 0 and 6x + 5y + 8 = 0.<\/p>\n

Solution : <\/span>The Given lines are x – y – 6 = 0 ….(i), 4x – 3y – 20 = 0 ….(ii) and 6x + 5y + 8 = 0 ….(iii)

\nfrom equation (i) and (ii), we get

\nx = 2 and y = -4

\nThus, the first two lines intersect at the point (2, -4). Putting x = 2 and y = -4 in equation (iii), we get

\n6\\(\\times\\)2 + 5\\(\\times\\)(-4) + 8 = 0

\nSo, Point (2, -4) lies on third line

\nHence, the given lines are concurrent and common point of intersection is (2, -4).<\/p>\n\n\n

Hope you learnt condition of concurrency of lines, learn more concepts of straight lines and practice more questions to get ahead in the competition. Good luck!<\/p>\n\n\n

\n
Next – Equation of Straight Lines in all Forms<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Orthocenter in a Triangle | Equation of Locus<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

There are two conditions of concurrency of lines which are given below : (a) Three lines are said to be concurrent if they pass through a common point i.e. they meet at a point. Thus, Three lines \\(a_1x + b_1y + c_1\\) = 0 and \\(a_2x + b_2y + c_2\\) = 0 and \\(a_3x + b_3y …<\/p>\n

Condition of Concurrency of Lines<\/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":[662,661],"yoast_head":"\nCondition of Concurrency of Lines - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn how to show that three lines are concurrent and its both condition to prove concurrency of lines 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\/concurrency-of-lines\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Condition of Concurrency of Lines - 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