Notes On Section Formula - CBSE Class 12 Maths

The position vector for R in both these cases.

Case I: R divides AB internally.

   OA   =  a        

   OB   =  b     

   OR   =  r    

Point R divide line segment AB in the ratio m:n

AR:RB = m:n

In ∆OAR:

OR    = OA   +  AR        

AR = OR - OA

AR = r - a

In ∆ORB:

OB = OR + RB

RB = OB - OR

RB = b - r

AR :RB = m:n

m RB = n AR

m (b - r ) = n ( r    - a   )

m b - m r = n r - n a

m r + n r = m b + n a

⇒ (m + n )r = m b + n a

r =  m b + n a m+n  

OR = m b + n a m+n  

The point R lies at the midpoint of line segment AB.

When m : n = 1:1

m = n

OR =   m b + n a m+n   =    m b + m a →  m+m  

OR =       m ( a + b ) 2m          

OR =   ( b + a ) 2   =  ( a + b ) 2       


Case II: R divides AB externally

   OA   =  a        

   OB   =  b     

   OR   =  r    

Let R divide AB externally such that AR:BR = m:n

In ∆OAR:

OR   = OA   + AR  

AR   = OR   - OA  

AR   = r   - a  

In ∆OBR:

OR   = OB   + BR  

BR   = OR   - OB  

BR   = r   - b  

AR   :BR   = m:n

m (BR  )= n (AR  )

m (r   - b  ) = n (r   - a   )

m r - m b = n r - n a

m r - n r = m b - n a

⇒ (m - n )r = m b - n a

r = m b - n a m-n  

OR = m b - n a m-n  .

Summary

The position vector for R in both these cases.

Case I: R divides AB internally.

   OA   =  a        

   OB   =  b     

   OR   =  r    

Point R divide line segment AB in the ratio m:n

AR:RB = m:n

In ∆OAR:

OR    = OA   +  AR        

AR = OR - OA

AR = r - a

In ∆ORB:

OB = OR + RB

RB = OB - OR

RB = b - r

AR :RB = m:n

m RB = n AR

m (b - r ) = n ( r    - a   )

m b - m r = n r - n a

m r + n r = m b + n a

⇒ (m + n )r = m b + n a

r =  m b + n a m+n  

OR = m b + n a m+n  

The point R lies at the midpoint of line segment AB.

When m : n = 1:1

m = n

OR =   m b + n a m+n   =    m b + m a →  m+m  

OR =       m ( a + b ) 2m          

OR =   ( b + a ) 2   =  ( a + b ) 2       


Case II: R divides AB externally

   OA   =  a        

   OB   =  b     

   OR   =  r    

Let R divide AB externally such that AR:BR = m:n

In ∆OAR:

OR   = OA   + AR  

AR   = OR   - OA  

AR   = r   - a  

In ∆OBR:

OR   = OB   + BR  

BR   = OR   - OB  

BR   = r   - b  

AR   :BR   = m:n

m (BR  )= n (AR  )

m (r   - b  ) = n (r   - a   )

m r - m b = n r - n a

m r - n r = m b - n a

⇒ (m - n )r = m b - n a

r = m b - n a m-n  

OR = m b - n a m-n  .

Videos

References

Previous
Next