JEE Main & Advanced Mathematics Rectangular Cartesian Coordinates Section Formulae

If \[P(x,y)\] divides the join of \[A({{x}_{1}},{{y}_{1}})\] and \[B({{x}_{2}},{{y}_{2}})\] in the ratio \[{{m}_{1}}:{{m}_{2}}({{m}_{1}},{{m}_{2}}>0)\]    

(1) Internal division :  If \[P(x,y)\] divides the segment AB internally in the ratio of \[{{m}_{1}}:{{m}_{2}}\]\[\Rightarrow \]\[\frac{PA}{PB}=\frac{{{m}_{1}}}{{{m}_{2}}}\]

The co-ordinates of  \[P(x,y)\] are

\[x=\frac{{{m}_{1}}{{x}_{2}}+{{m}_{2}}{{x}_{1}}}{{{m}_{1}}+{{m}_{2}}}\] and \[y=\frac{{{m}_{1}}{{y}_{2}}+{{m}_{2}}{{y}_{1}}}{{{m}_{1}}+{{m}_{2}}}\]

(2) External division :  If \[P(x,y)\]divides the segment AB externally in the ratio of \[{{m}_{1}}:{{m}_{2}}\]\[\Rightarrow \]\[\frac{PA}{PB}=\frac{{{m}_{1}}}{{{m}_{2}}}\]\[\]

The co-ordinates of \[P(x,y)\] are    

\[x=\frac{{{m}_{1}}{{x}_{2}}-{{m}_{2}}{{x}_{1}}}{{{m}_{1}}-{{m}_{2}}}\] and \[y=\frac{{{m}_{1}}{{y}_{2}}-{{m}_{2}}{{y}_{1}}}{{{m}_{1}}-{{m}_{2}}}\]