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
L2
L3
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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