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

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