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Point of Intersection of Two Lines

We will learn how to find the co-ordinates of the point of intersection of two lines.

Let the equations of two intersecting straight lines be

a1 x + b1y + c1  = 0 ………….. (i) and   

a2 x + b2 y + c2 = 0 …….…... (ii)

Suppose the above equations of two intersecting lines intersect at P(x1, y1). Then (x1, y1) will satisfy both the equations (i) and (ii).

Therefore, a1x1 + b1y1  + c1 = 0 and

a2x1 + b2y1 + c2 = 0      

Solving the above two equations by using the method of cross-multiplication, we get,

x1b1c2βˆ’b2c1=y1c1a2βˆ’c2a1=1a1b2βˆ’a2b1

Therefore, x1  = b1c2βˆ’b2c1a1b2βˆ’a2b1 and

y1  = c1a2βˆ’c2a1a1b2βˆ’a2b1,  a1b2 - a2b1 β‰  0

Therefore, the required co-ordinates of the point of intersection of the lines (i) and (ii) are

(b1c2βˆ’b2c1a1b2βˆ’a2b1, (c1a2βˆ’c2a1a1b2βˆ’a2b1), a1b2 - a2b1 β‰  0


Notes: 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.

If a1b2 - a2b1 = 0 then a1b2 = a2b1

β‡’ a1b1 = a2b2

β‡’ - a1b1 = - a2b2  i.e., the slope of line (i) = the slope of  line  (ii)

Therefore, in this case the straight lines (i) and (ii) are parallel and hence they do not intersect at any real point.


Solved example to find the co-ordinates of the point of intersection of two given intersecting straight lines:

Find the coordinates of the point of intersection of the lines 2x - y + 3 = 0 and x + 2y - 4 = 0.

Solution:

We know that the co-ordinates of the point of intersection of the lines a1 x+ b1y+ c1  = 0 and a2 x + b2 y + c2 = 0 are

(b1c2βˆ’b2c1a1b2βˆ’a2b1, (c1a2βˆ’c2a1a1b2βˆ’a2b1), a1b2 - a2b1 β‰  0

Given equations are

2x - y + 3 = 0 …………………….. (i)

x + 2y - 4 = 0 …………………….. (ii)

Here a1 = 2, b1 = -1, c1 = 3, a2 = 1, b2 = 2 and c2 = -4.

((βˆ’1)β‹…(βˆ’4)βˆ’(2)β‹…(3)(2)β‹…(2)βˆ’(1)β‹…(βˆ’1), (3)β‹…(1)βˆ’(βˆ’4)β‹…(2)(2)β‹…(2)βˆ’(1)β‹…(βˆ’1))

β‡’ (4βˆ’64+1, 3+84+1)

β‡’ (115,βˆ’25)

Therefore, the co-ordinates of the point of intersection of the lines 2x - y + 3 = 0 and x + 2y - 4 = 0 are (115,βˆ’25).

● The Straight Line





11 and 12 Grade Math

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