"CONCURRENCY OF THREE LINES"

 

“Concurrency of Three Lines”

When two or more straight lines meet in a point,they are said to be “Concurrent” and their meeting point is called the “Point of Concurrency”.

     Where d is the point of concurrency.

Let the equations of the three intersecting lines l₁, l₂ and l₃ be respectively.

(i)     l₁: a₁x+b₁y+c₁=0

(ii)    l₂:a₂x+b₂y+c₂=0

(iii)   l₃: a₃x+b₃y+c₃=0

If the above lines are concurrent then condition of concurrency for lines is written in the determinant form as:                                

                                                      

      

      
The above condition is not sufficient to ensure that the three given lines are concurrent .However, it can be shown that, if the above determinant vanishes, then either the given lines are parallel or concurrent.

Here one has to investigate the coefficients; if coefficients of the variables are not same then lines are concurrent.

problem

Q:  Show that the following lines are concurrent. Also find their point of concurrency.

(i) x-y = 6, 4y +22 = 3x  and  6x+5y+8= 0

solution:

                                        i.         l₁ : x-y = 6…………..eq(i)

                                      ii.         l₂ : 4y+22 = 3x………eq(ii)

                                    iii.         l₃ : 6x+5y+8 = 0…….eq(iii)

 Therefore equation (i) will be;

ð x= 6+y……..(A)

Putting in eq (ii); we get;

ð 4y+22 = 3(6+y)

ð 4y+22 =18+3y

ð 4y-3y= 18-22

ð y = -4

Putting in eq (A); we get;

ð x= 6+(-4)

ð x=2

Therefore   (2,-4) are the coordinates of point of intersection of l₁ and l₂. If line 

l₃: 6x+5y+8=0 also passes through the same  point then:

ð 6(2)+5(-4)+8 = 0

ð 12-20+8=0

ð -8+8=0---------àwhich is true.

Therefore it is proved that lines are concurrent and (2,-4) is the point of concurrency.