# 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: $Ax+By+{C}_{1}=0$
• Line 2: $Px+Qy+{C}_{2}=0$
• Line 3: $Mx+Ny+{C}_{3}=0$
• 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. $2x-3y+4=0$
2. $4x+y-6=0$
3. $6x-2y+8=0$

### 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: $2x-3y+4=0$
• Equation 2: $4x+y-6=0$

Solving these equations simultaneously gives:

1. $2x-3y+4=0$ ...(1)
2. $4x+y-6=0$ ...(2)

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

$12x+3y-18=0$ ...(3)

$14x-14=0$
$14x=14$
$x=1$

Substitute $x=1$ into equation (1):

$2\left(1\right)-3y+4=0$
$2-3y+4=0$
$-3y+6=0$
$-3y=-6$
$y=2$

Hence, the point of intersection for Line 1 and Line 2 is $\left(1,2\right)$.

#### Step 2: Verifying Line 3:

• Equation 3: $6x-2y+8=0$
• Substitute the obtained intersection point $\left(1,2\right)$ into the equation of Line 3:

$6\left(1\right)-2\left(2\right)+8=6-4+8=10\mathrm{\ne }0$

### Conclusion:

As the coordinates $\left(1,2\right)$ do not satisfy the equation of Line 3, the lines $2x-3y+4=0$, $4x+y-6=0$, and $6x-2y+8=0$ 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.