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 slopeintercept 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 slopeintercept 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 yintercepts 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
11 and 12 Grade Math
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