If $A (x_1,y_1)$ and $B (x_2 , y_2)$ be the coordinates of $\overline{AB}. P (x,y)$ divides $\overline{AB}$ in the ratio $m : n$ (Either internally or externally). Then it can be expressed as $\dfrac{m}{n}: 1$ or $\lambda : 1 \left( \lambda =\dfrac{m}{n} \right)$ where,

$\to$ For Internal Division:

$x=\dfrac{\lambda x_2+ x_1}{\lambda+1} , \quad y=\dfrac{\lambda y_2+y_1}{\lambda+1}$

$\to$ For External Division

$x=\dfrac{\lambda x_2- x_1}{\lambda-1} , \quad y=\dfrac{\lambda y_2-y_1}{\lambda-1}$

(Hence $\lambda \neq 1$ )

Some Example on the above Properties:

$\to$ Distance Formula

Coordinates of $P$ and $Q$ are $(3,5)$ and $(2,-7)$ respectively, then

$\begin{aligned} PQ&=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\&=\sqrt{(3-2)^2+\{5-(-7)\}^2}\\&=\sqrt{1^2+12^2}\\&=\sqrt{1+144}\\&=\sqrt{145}\end{aligned}$

$\to$ Distance from Origin $0 (0,0)$

Let $P (3,-8)$ be a point on a Cartesian Plane then

$\begin{aligned} OP&=\sqrt{3^2+(-8)^2}\\&=\sqrt{9+64}\\&=\sqrt{73}\end{aligned}$

$\to$ Internal Division on Ratio $(m : n )$

Let $A (4,9)$ and $B (7,-5)$ be the end points of a line segment $AB$ . $P$ divides $AB$ internally in ratio $2 : 3$ , the coordinate of $P (x,y)$ is

$\begin{aligned} x&=\dfrac{mx_2+nx_1}{m+n}, \quad y=\dfrac{my_2+ny_1}{m+n}\\&=\dfrac{2\times7+3\times 7}{2+3},\quad =\dfrac{2\times (-5)+3\times 9}{2+3}\\&=\dfrac{14+21}{5},\quad\quad\quad =\dfrac{-10+27}{5}\\&=\dfrac{35}{5}=7,\qquad \quad\quad=\dfrac{17}{5}\end{aligned}$

$\therefore P(x,y) =\left( 7,\dfrac{17}{5} \right)$

Internal Division in Ratio $(\lambda : 1 )$

Let $A (7,-3)$ and $B (-4,9)$ be the end points of $\overline{AB}$ . $P$ divides $\overline{AB}$ in ratio $3 : 1$ internally the coordinate of $P$ is

$\begin{aligned} x&=\dfrac{\lambda x_2+x_1}{\lambda +1}, \qquad y=\dfrac{\lambda y_2 +y_1}{\lambda +1}\\&=\dfrac{3(-4)+7}{3+1},\quad =\dfrac{3\times 9+(-3)}{3+1}\\&=\dfrac{-12+7}{4},\quad\quad\quad =\dfrac{27-3}{4}\\&=\dfrac{-5}{4}\qquad \quad \quad\quad=6\end{aligned}$

$\therefore P(x,y) =P \left( -\dfrac{5}{4},6 \right)$

External Division in ratio $(m : n)$

Let $A(2,3)$ and $B (7,9)$ be two end points on $\overline{AB}$ . Let $P (x,y)$ be a point divides $\overline{AB}$ externally in ratio $5 : 3$ , then

$\begin{aligned} x&=\dfrac{mx_2-nx_1}{m-n}, \qquad y=\dfrac{my_2-ny_1}{m-n}\\&=\dfrac{5\times 7-3\times2}{5-3},\quad =\dfrac{5\times9-3 \times 3}{5-3}\\&=\dfrac{35-6}{2},\quad\quad\quad =\dfrac{45-9}{2}\\&=\dfrac{29}{2}=14.5, \quad\quad=17\end{aligned}$

$\therefore P (x,y) = P ( 14.5,17)$

External Division in Ratio $( \lambda : 1)$

Let $A (5,-3)$ and $B (11, 13)$ be the two end points of $\overline{AB}$ . Let $P (x,y)$ be a point divides $\overline{AB}$ in ratio $4:1$ , externally, then

$\begin{aligned} x&=\dfrac{\lambda x_2 -x_1}{\lambda-1}, \qquad y=\dfrac{\lambda y_2-y_1}{\lambda-1}\\&=\dfrac{4\times 11-5}{4-1},\quad =\dfrac{4\times 13-(-3)}{4-1}\\&=\dfrac{44-5}{3},\quad\quad\quad =\dfrac{52+3}{3}\\&=\dfrac{39}{3}, \quad\quad=\dfrac{55}{3}\\&=13\end{aligned}$

Mid-Point Formula:

Let $A (5,9)$ and $B (7,-3)$ be end points of $\overline {AB}$ and $P (x,y)$ be midpoint of $\overline {AB}$ , then

$\begin{aligned} x&=\dfrac{ x_1+x_2}{2}, \qquad y=\dfrac{ y_1+y_2}{2}\\&=\dfrac{5+7}{2},\quad =\dfrac{9+(-3)}{2}\\&=6,\quad\quad\quad =3\end{aligned}$

$\therefore P(x,y) = P(6,3)$

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