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

a\(_{1}\) x + b\(_{1}\)y + c\(_{1}\)  = 0 ………….. (i) and   

a\(_{2}\) x + b\(_{2}\) y + c\(_{2}\) = 0 …….…... (ii)

Suppose the above equations of two intersecting lines intersect at P(x\(_{1}\), y\(_{1}\)). Then (x\(_{1}\), y\(_{1}\)) will satisfy both the equations (i) and (ii).

Therefore, a\(_{1}\)x\(_{1}\) + b\(_{1}\)y\(_{1}\)  + c\(_{1}\) = 0 and

a\(_{2}\)x\(_{1}\) + b\(_{2}\)y\(_{1}\) + c\(_{2}\) = 0      

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

\(\frac{x_{1}}{b_{1}c_{2} - b_{2}c_{1}} = \frac{y_{1}}{c_{1}a_{2} - c_{2}a_{1}} = \frac{1}{a_{1}b_{2} - a_{2}b_{1}}\)

Therefore, x\(_{1}\)  = \(\frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}\) and

y\(_{1}\)  = \(\frac{c_{1}a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}\),  a\(_{1}\)b\(_{2}\) - a\(_{2}\)b\(_{1}\) ≠ 0

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

(\(\frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}\), (\(\frac{c_{1}a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}\)), a\(_{1}\)b\(_{2}\) - a\(_{2}\)b\(_{1}\) ≠ 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 a\(_{1}\)b\(_{2}\) - a\(_{2}\)b\(_{1}\) = 0 then a\(_{1}\)b\(_{2}\) = a\(_{2}\)b\(_{1}\)

⇒ \(\frac{a_{1}}{b_{1}}\) = \(\frac{a_{2}}{b_{2}}\)

⇒ - \(\frac{a_{1}}{b_{1}}\) = - \(\frac{a_{2}}{b_{2}}\)  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 a\(_{1}\) x+ b\(_{1}\)y+ c\(_{1}\)  = 0 and a\(_{2}\) x + b\(_{2}\) y + c\(_{2}\) = 0 are

(\(\frac{b_{1}c_{2} - b_{2}c_{1}}{a_{1}b_{2} - a_{2}b_{1}}\), (\(\frac{c_{1}a_{2} - c_{2}a_{1}}{a_{1}b_{2} - a_{2}b_{1}}\)), a\(_{1}\)b\(_{2}\) - a\(_{2}\)b\(_{1}\) ≠ 0

Given equations are

2x - y + 3 = 0 …………………….. (i)

x + 2y - 4 = 0 …………………….. (ii)

Here a\(_{1}\) = 2, b\(_{1}\) = -1, c\(_{1}\) = 3, a\(_{2}\) = 1, b\(_{2}\) = 2 and c\(_{2}\) = -4.

(\(\frac{(-1)\cdot (-4) - (2)\cdot (3)}{(2)\cdot (2) - (1)\cdot (-1)}\), \(\frac{(3)\cdot (1) - (-4)\cdot (2)}{(2)\cdot (2) - (1)\cdot (-1)}\))

⇒ (\(\frac{4 - 6}{4 + 1}\), \(\frac{3 + 8}{4 + 1}\))

⇒ (\(\frac{11}{5}, \frac{-2}{5}\))

Therefore, the co-ordinates of the point of intersection of the lines 2x - y + 3 = 0 and x + 2y - 4 = 0 are (\(\frac{11}{5}, \frac{-2}{5}\)).

● The Straight Line

• Straight Line
• Slope of a Straight Line
• Slope of a Line through Two Given Points
• Collinearity of Three Points
• Equation of a Line Parallel to x-axis
• Equation of a Line Parallel to y-axis
• Slope-intercept Form
• Point-slope Form
• Straight line in Two-point Form
• Straight Line in Intercept Form
• Straight Line in Normal Form
• General Form into Slope-intercept Form
• General Form into Intercept Form
• General Form into Normal Form
  
• Point of Intersection of Two Lines
  
• Concurrency of Three Lines
• Angle between Two Straight Lines
• Condition of Parallelism of Lines
• Equation of a Line Parallel to a Line
• Condition of Perpendicularity of Two Lines
• Equation of a Line Perpendicular to a Line
• Identical Straight Lines
• Position of a Point Relative to a Line
• Distance of a Point from a Straight Line
• Equations of the Bisectors of the Angles between Two Straight Lines
• Bisector of the Angle which Contains the Origin
• Straight Line Formulae
• Problems on Straight Lines
• Word Problems on Straight Lines
• Problems on Slope and Intercept

11 and 12 Grade Math

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