Geo2.24

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If the equations a_{1}x+b_{1}y+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}\neq 0,a_{2}\neq 0\,

Then the given equations are

a_{1}x+c_{1}=0,a_{2}x+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}\neq 0,b_{2}\neq 0\,

Hence,the equation

a_{1}x+b_{1}y+c=0\, can be written as

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

and

a_{2}x+b_{2}y+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}}}\,


Main Page:Geometry:Straight Lines-I