Identical Straight Lines

When the coefficients of two straight lines are proportional they are called identical straight lines.

Let us assume, the straight lines a\(_{1}\) x + b\(_{1}\) y + c\(_{1}\) = 0 and a\(_{2}\) x + b\(_{2}\)y + c\(_{2}\)  = 0 are identical then

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

To get the clear concept let us proof the above statement:

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

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

Convert the straight line a\(_{1}\)x + b\(_{1}\)y + c\(_{1}\) = 0 in slope-intercept form we get,

y = \(\frac{a_{1}}{b_{1}}\)x - \(\frac{c_{1}}{b_{1}}\)

Similarly, convert the straight line a\(_{2}\)x + b\(_{2}\)y + c\(_{2}\) = 0 in slope-intercept form we get,

y = \(\frac{a_{2}}{b_{2}}\)x - \(\frac{c_{2}}{b_{2}}\)

If (i) and (ii) represent the equations of the same straight line then their slopes are equal.

i.e., - \(\frac{a_{1}}{b_{1}}\) = - \(\frac{a_{2}}{b_{2}}\)

or, \(\frac{a_{1}}{a_{2}}\) = \(\frac{b_{1}}{b_{2}}\) .…………………..(iii)

Again, the y-intercepts of lines (i) and (ii) are also equal.

Therefore,  - \(\frac{c_{1}}{b_{1}}\) = - \(\frac{c_{2}}{b_{2}}\)

or, \(\frac{b_{1}}{b_{2}}\) = \(\frac{c_{1}}{c_{2}}\) .…………………..(iv)

Therefore, from (iii) and (iv) it is clear that (i) and (ii) will represent the same straight line when

\(\frac{a_{1}}{a_{2}}\) = \(\frac{b_{1}}{b_{2}}\) = \(\frac{c_{1}}{c_{2}}\).

● 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

From Identical Straight Lines to HOME PAGE