If $A-P-B$ , the $P$ is said to be divide the line segment $AB$ internally in the ratio $m:n$ , where coordinates of $A , B$ , and $P$ are $(x_1,y_1), (x_2,y_2)$ and $(x,y)$ respectively, and $\dfrac{PA}{PB}=\dfrac{m}{n}$ , then $x=\dfrac{mx_2+nx_1}{m+n}, \quad y=\dfrac{my_2+ny_1}{m+n}$

Hence, $PA+PB=AB$

(ii) External Division:

If $P-A-B$ or $A-B-P$ , then $P$ is said to be divide the line segment $\overline{AB}$ externally into the segments $\overline{PA}$ and $\overline{PB}$ .

The ratio of external division being given by either $PA: PB$ or $PB : PA$ .

Hence $|PA-PB|=AB$

Case (1)

Hence $PB-PA=AB$

Case (2)

Hence $PA-PB =AB$

Formula:

If $P (x,y)$ divides $\overline{AB}$ , the line segment joining $A(x_1,y_1)$ and $B (x_2, y_2)$ externally where $\dfrac{PA}{PB}=\dfrac{m}{n}$ , then $x=\dfrac{mx_2-nx_1}{m-n}, \quad y=\dfrac{my_2-ny_1}{m-n}$

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