Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Mid Point Formula

If  P(x,y) is the midpoint of the line joining  A (x_1 , y_1) and  B (x_2, y_2) then 

 x=\dfrac{x_1+x_2}{2}, \quad y=\dfrac{y_1+y_2}{2}

Hence  PA:PB = 1 : 1

 P(x,y)=\left( \dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2} \right)

Convention:

In case of internal division:

 M is the midpoint of  \overline{AB} .

Hence  P  be a point on  \overline{AB} , which divides  \overline{AB} internally.

There are two conditions:

(i)  \dfrac{PA}{PB} < 1\quad \text { or }\quad \dfrac{PB}{PA} > 1

(ii)  \dfrac{PA}{PB} > 1\quad \text { or }\quad \dfrac{PB}{PA} < 1

 Note:

(i) We write 'P' divides  \overline{AB} or the line segment joining   A and  B in ratio  m : n to mean that \dfrac{PA}{PB}=\dfrac{m}{n} .

(ii) For 'P' divides  \overline{BA} or the line segment joining   B and  A in ratio  m : n to mean that  \dfrac{PB}{PA}=\dfrac{m}{n} .

In Case of External Division:

 \dfrac{PA}{PB} < 1 \implies P-A-B

and  \dfrac{PA}{PB} > 1 \implies A-B-P

Alternatively, 

\dfrac{PB}{PA} > 1 \implies P-A-B

and \dfrac{PB}{PA} < 1 \implies A-B-P

Coordinates of any Points on the line:

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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