# 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 a1x + b1y + c1 = 0

L2 a2x + b2y + c2 = 0

L3 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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