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

Division Formula

(I) Internal Division

    If   P (x, y, z)  divides the line segment joining  A (x_1,y_1, z_1) and  B (x_2,y_2,z_2) internally in ratio  m:n i.e.

 \dfrac{PA}{PB}-\dfrac{m}{n}, then,

 x=\dfrac{mx_2+nx_1}{m+n}, y=\dfrac{my_2 +ny_1}{m+n}, z=\dfrac{mz_2+n_2}{m+n}  

Example:

Let  A (2, 4, -5) and  B (3, 4, 5) be two point.  P (x, y, z) divides  \overline{AB} internally in ratio 2:3 then

 x=\dfrac{mx_2+nx_1}{m+n)}=\dfrac{2\times 3+3\times2}{2+3}=\dfrac{12}{5}

 y=\dfrac{my_2+ny_1}{m+n}=\dfrac{2\times 4+3\times4}{2+3}=\dfrac{20}{5}=4

 z=\dfrac{mz_2+nz_1}{m+n}=\dfrac{2\times 5+3\times(-5)}{2+3}=\dfrac{-5}{5}=-1

\therefore P (x, y, z)=(\dfrac{12}{5}, 4, -1)

(II) External Division

    If  P(x, y, z) divides  \overline{AB} joining  A (x_1, y_1, z_1) and   B (x_2,y_2,z_2) externally in ratio  m:n i.e.  \dfrac{PA}{PB}=\dfrac{m}{n}, then

 x=\dfrac{mx_2-nx_1}{m-n}, \quad y= \frac{my_2-ny_1}{m-n} \quad z= \dfrac{mz_2-nz_1}{m-n}

\to It has two cases:

(a) For  m > n,  \dfrac{PA}{PB} > 1 i.e.  A-B-P

(b) For  m< n, \dfrac{PA}{PB} < 1 i.e.  P-A-B

Example: 

Let  A( 7, 5, -2) and  B(3, 1, 4) be two points.  P(x, y, z) divides  \overline{AB} externally in ratio  5:3 then

  \begin{aligned} x&= \dfrac{mx_2-nx_1}{m-n}= \dfrac{5 \times3-3\times7}{5-3}=\dfrac{15-21}{2} \\&=\dfrac{-6}{2}=-3 \end{aligned}

 \begin{aligned} y&= \frac{my_2-ny_1}{m-n}= \frac{5 \times1-3\times5}{5-3}=\frac{5-15}{2}\\&=\frac{-10}{2}=-5 \end{aligned}

\begin{aligned} z&= \dfrac{mz_2-nz_1}{m-n}= \dfrac{5 \times4-3(-2)}{5-3}=\dfrac{20+6}{2} \\&=\frac{26}{2}=13\end{aligned}

 \therefore P(x, y, z) = (-3,-5, 13)

Midpoint Formula

If  M(x, y, z)   be midpoint of  \overline{AB} joining  A (x_1, y_1, z_1)  and  B (x_2,y_2,z_2),  then

 x=\dfrac{x_1+x_2}{2}, \quad y=\dfrac{y_1+y_2}{2}, \quad z=\dfrac{z_1+z_2}{2}, \quad

Example:

Let  A(3, 7, 2) and  B(7, -5, 6) be two points in space and  M(x, y, z) be midpoint of   \overline{AB} , then

 x=\dfrac{x_1+x_1}{2}=\dfrac{3+7}{2}=5

 y=\dfrac{7+(-5)}{2}=2

 z=\dfrac{z_1+z_2}{2}=\frac{2+6}{2}=4

\therefore  M(x, y, z)=(5, 2, 4)

Division Formula by  \lambda :1 ratio

If  P divides  \overline{AB} in ratio  \lambda :1 then coordinates of  P  are given by:

(i)  x=\dfrac{\lambda x_2+x_1}{\lambda+1}, \quad y=\dfrac{\lambda y_2+ y_1}{\lambda+1}, \quad z=\dfrac{\lambda z_2+z_1}{\lambda+1} (Internal Division)

(ii)  x=\dfrac{\lambda x_2-x_1}{\lambda-1}, \quad y=\dfrac{\lambda y_2- y_1}{\lambda-1}, \quad z=\dfrac{\lambda z_2-z_1}{\lambda-1} (External Division)

Note:

If  A (x_1,y_1,z_1) and  B (x_2,y_2,z_2) are distinct points then the coordinates of any point on  \overleftrightarrow{AB} except  A  \text { and }B are given by

 \left( \dfrac{x_1+\mu x_2}{1+\mu}, \dfrac{y_1+\mu y_2}{1+ \mu},\dfrac{z_1+\mu z_2}{1+\mu} \right)

 \mu \in R ,\quad \mu \neq -1

Formulas to find Coordinates

If  A(x_1, y_1, z_1) ,  B(x_2,y_2,z_2) and   C(x_3,y_3,z_3) are vertices of triangle  ABC  , whose sides  BC, CA, AB are of lengths  a,b,c respectively, then

(i) Centroid (G)=\left( \dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3},\dfrac{z_1+z_2+z_3}{3} \right)

(ii) Incentre (I)=\left( \dfrac{ax_1+bx_2+cx_3}{a+b+c}, \dfrac{ay_1+by_2+cy_3}{a+b+c},\dfrac{az_1+bz_2+cz_3}{a+b+c}\right)

(iii) Excentre 

(a) (to A) (I_1)=\left( \dfrac{-ax_1+bx_2+cx_3}{-a+b+c}, \dfrac{-ay_1+by_2+cy_3}{-a+b+c},\dfrac{-az_1+bz_2+cz_3}{-a+b+c}\right)

(b) (to B) (I_2)=\left( \dfrac{ax_1-bx_2+cx_3}{a-b+c}, \dfrac{ay_1-by_2+cy_3}{a-b+c},\dfrac{az_1-bz_2+cz_3}{a-b+c}\right)

(c) (to C)  (I_3)=\left( \dfrac{ax_1+bx_2-cx_3}{a+b-c}, \dfrac{ay_1+by_2-cy_3}{a+b-c},\dfrac{az_1+bz_2-cz_3}{a+b-c}\right)

\to If  A,B,C are vertices of triangle ABC and  P(x_x, y_1, z_1),   Q(x_2,y_2,z_2), and  R(x_3,y_3,z_3),  are coordinates of midpoint of  \overline{BC},\overline{AC} \text { and } \overline{AB} respectively,then coordinate of Centroid of (G)=\left( \dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3},\dfrac{z_1+z_2+z_3}{3}\right)

 

 

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