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
\(\frac{a_{1}}{a_{2}}\) = \(\frac{b_{1}}{b_{2}}\) = \(\frac{c_{1}}{c_{2}}\).
11 and 12 Grade Math
From Identical Straight Lines to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
