In this page we are going to see the definition of concurrency of straight lines.
Definition:
If three or more straight lines passing through the same point then that common point is called the point of concurrency
Steps to find whether the three given lines are concurrent:
(i) Solve any two equations of the straight lines and obtain their point of intersection.
(ii) Substitute the co-ordinates of the point of intersection in the third equation.
(iii) Check whether the third equation is satisfied
(iv) If it is satisfied,the point lies on the third line and so the three straight lines are concurrent.
Example 1:
Show that the straight lines 2x - 3y +4 = 0,9x + 5y = 19 and
2x -7y + 12 = 0 are concurrent.Find the point of concurrency.
Solution:
The given equations are
2x - 3y + 4 = 0 ----------(1)
9x + 5y = 19 ----------(2)
2x -7y + 12 = 0 ----------(3)
First we need to solve any two equations then we have to plug the point into the another equation
(1) x 5 => 10x - 15y = -20
(2) x 3 => 27x + 15y = 57
----------------
37x = 37
x = 37/37
x = 1
Substituent x = 1 in the first equation
2(1) - 3y = -4
2 - 3y = -4
-3y = -4-2
-3y = -6
y = -6/(-3)
y = 2
So the point of intersection of the first and second line is (1,2)
Now we have to apply the point (1,2) in the third equation 2x -7y + 12 = 0
2(1) - 7(2) + 12 = 0
2 -14 +12 = 0
-12 + 12 = 0
0 = 0
The third equation is satisfied.So the point (1,2) lies on the lies on the third line.So the straight lines are concurrent. concurrency of straight lines
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