Condition of Concurrency of Straight Lines

Introduction:

  • Concurrency of lines refers to the condition where three or more lines intersect at a single point.

Condition for Concurrency:

  • In a Cartesian plane, three lines are concurrent if and only if the point of intersection of any two lines also lies on the third line.

Theorem of Concurrency:

  • Three lines represented by equations:
    • Line 1: Ax+By+C1=0
    • Line 2: Px+Qy+C2=0
    • Line 3: Mx+Ny+C3=0
  • Condition for Concurrency:
    • If the point of intersection of Line 1 and Line 2 also lies on Line 3 (or vice versa), then the three lines are concurrent.

Method to Check Concurrency:

  • Step 1: Find Intersections:
    • Find the point of intersection between Line 1 and Line 2 using simultaneous equations.
  • Step 2: Verify the Third Line:
    • Substitute the obtained coordinates into the equation of the third line.
    • If the coordinates satisfy the equation, the lines are concurrent.

Example:

Given three lines:

  1. 2x3y+4=0
  2. 4x+y6=0
  3. 6x2y+8=0

Checking for Concurrency:

Step 1: Finding Intersection of Line 1 and Line 2:

  • Solve equations 1 and 2 to find the point of intersection.
  • Equation 1: 2x3y+4=0
  • Equation 2: 4x+y6=0

Solving these equations simultaneously gives:

  1. 2x3y+4=0 ...(1)
  2. 4x+y6=0 ...(2)

Multiply equation (2) by 3 to eliminate 'y':

12x+3y18=0 ...(3)

Add equations (1) and (3):

14x14=0
14x=14
x=1

Substitute x=1 into equation (1):

2(1)3y+4=0
23y+4=0
3y+6=0
3y=6
y=2

Hence, the point of intersection for Line 1 and Line 2 is (1,2).

Step 2: Verifying Line 3:

  • Equation 3: 6x2y+8=0
  • Substitute the obtained intersection point (1,2) into the equation of Line 3:

6(1)2(2)+8=64+8=100

Conclusion:

As the coordinates (1,2) do not satisfy the equation of Line 3, the lines 2x3y+4=0, 4x+y6=0, and 6x2y+8=0 are not concurrent.

Importance:

  • Concurrency of lines signifies a significant intersection point where multiple relationships and geometrical properties can be derived.
  • It's crucial in various fields such as architecture, engineering, and mathematics to understand the intersection and relationships between multiple lines.