Geo2.24

If the equations $a_1x+b_1y+c=0,a_2+b_2+c_2=0\,$ represent the same straight line then prove that $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\,$

If the given equations represent a vertical line, then

$b_1=0,b_2=0,a_1\ne 0,a_2\ne 0\,$

Then the given equations are

$a_1x+c_1=0,a_2x+c_2=0\,$

Eliminating x from these equations

$\frac{a_1}{a_2}=\frac{c_1}{c_2}\,$

If the given equations represent a non-vertical line,then

$b_1\ne 0,b_2\ne 0\,$

Hence,the equation

$a_1x+b_1y+c=0\,$ can be written as

$y=\frac{-a}{b}x-\frac{c}{b}\,$

and

$a_2x+b_2y+c_2=0\,$

can be written as

$y=\frac{-a_2}{b_2}x-\frac{c_2}{b_2}\,$

Since both the above equations represents the same line,their slopes are equal.

$\frac{-a_1}{b_1}=\frac{-a_2}{b_2},\frac{-c_1}{b_1}=\frac{-c_2}{b_2}\,$

Hence

$\frac{a_1}{a_2}=\frac{b_1}{b_2},\frac{b_1}{b_2}=\frac{c_1}{c_2}\,$

$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}\,$