{"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 (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) 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
\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\nAnother Condition of Concurrency of Lines :<\/h2>\n
\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