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Point of intersection and concurrency of lines


Point of Intersection of Two Lines

Let the equations of two lines be
a1x + b1y + c1 = 0 – (i)
a2x + b2y + c2 = 0 – (ii)
Suppose these two lines intersect at a point P(x1, y1). Then, (x1, y1) satisfies each of the given equations.
Therefore, a1x1 + b1y1 + c1 = 0 and a2x¬1 + b2y1 + c2 = 0
Solving these two by cross multiplication, we get
       x1       =        y1         =       1
b1c2 – b2c1       c1a2 – c2a1      a1b2 – a2b1
x1 = b1c2 – b2c1
       a1b2 – a2b1
y1 = c1a2 – c2a1     
        a1b2 – a2b1

Hence, the coordinates of the point of intersection of (i) and (ii) are:

b1c2 – b2c1   , c1a2 – c2a1     
a1b2 – a2b1       a1b2 – a2b1

Important Remark

1. To find the coordinates of the point of intersection of two non – parallel lines, we solve the given equations simultaneously and the values of x and y so obtained determine the coordinates of the point of intersection.

Condition of Concurrency of three lines

Three lines are said to be concurrent if they pass through a common point i.e. they meet at a point.
Thus, if three lines are concurrent the point of intersection of two lines lies on the third line. Let
a1x + b1y + c1 = 0 – (i)
a2x + b2y + c2 = 0 – (ii)
a3x + b3y + c3 = 0 – (iii)
be three concurrent lines.

Then the point of intersection of (i) and (ii) must lie on the third. The coordinates of the point of intersection of (i) and (ii) are

b1c2 – b¬2c1,    c1a2 – c2a1 
a1b2 – a2b1     a1b2 – a2b1
This point must lie on (iii)

Therefore, a3 b1c2 – b¬2c1 + b3 c1a2 – c2a1   + c3 = 0

                   a1b2 – a2b1          a1b2 – a2b1
a3(b1c2 – b2c1) + b3(c1a2 – c2a1) + c3(a1b2 – a2b1) = 0

a1  b1  c1
a2  b2  c2   = 0
a3  b3  c3
This is the required condition of concurrency of three lines.

Another condition of concurrency of three lines

Three lines:
L1 square a1x + b1y + c1 = 0
L2 square a2x + b2y + c2 = 0
L3 square a3x + b3y + c3 = 0
are concurrent iff there exist constants λ1, λ2, λ3 not all zero such that
λ1L1 + λ2L2 + λ3L3 = 0
λ1 (a1x + b1y + c1) + λ2 (a2x + b2y + c2) + λ3 (a3x + b3y + c3) = 0

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