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The ratio in which the straight line L=ax+by+c=0\,divides the line joining the points A(x_{1},y_{1}),B(x_{2},y_{2})\, is {\frac  {-L_{[}x_{1},y_{1}]}{L_{[}x_{2},y_{2}]}}\,.

Let the straight line L=ax+by+c=0 divide the line joining A(x_{1},y_{1})B(x_{2},y_{2})\,

in the ratio l:m at P.

Since AP:PB=l:m

P=\left({\frac  {lx_{2}+mx_{1}}{l+m}},{\frac  {ly_{2}+my_{1}}{l+m}}\right)\,

Point P lies on L=0, hence

a[{\frac  {lx_{2}+mx_{1}}{l+m}}]+b[{\frac  {ly_{2}+my_{1}}{l+m}}+c=0\,

Readjusting the variables

{\frac  {l}{m}}={\frac  {-(ax_{1}+by_{1}+c)}{ax_{2}+by_{2}+c}}={\frac  {-L_{(}x_{1},y_{1})}{L_{(}x_{2},y_{2})}}\,

Main Page:Geometry:Straight Lines-I