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

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

11 and 12 Grade Math

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