MathematicsIf a point P(2,1)   is located on the line segment connecting two points A(4,2)   and B(8,4),   then from the following choices, find the correct one.

If a point P(2,1)   is located on the line segment connecting two points A(4,2)   and B(8,4),   then from the following choices, find the correct one.


  1. A
    AP= 1 2 AB  
  2. B
    AB= 1 2 AP  
  3. C
    AP= 1 4 AB  
  4. D
    AP= 1 3 AB   

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    Solution:

    Given that, the points are A(4, 2) and B(8, 4) and P(2,1)   is located on the line segment.
    Using the section formula, if a point ( x,y  ) divides the line joining the points ( x 1 , y 1  ) and (x2,y2) in the ratio  m:n  , then
    x,y =( m x 2 +n x 1 m+n , m y 2 +n y 1 m+n )  .
    Let  P   divides  AB   in the ratio  k:1  .
    Hence, m = k, n = 1.
    Here,
    ( x 1 , y 1 )=(4,2) ( x 2 , y 2 )=(8,4)  
    Substitute the values in the formula, we get,
    2,1 =( 8k+4 k+1 , 4k+2 k+1 )   Comparing the x-coordinate,
    8k+4 k+1 =2   8k+4=2(k+1) 8k+4=2k+2 8k2k=4+2 6k=2 k= 1 3  
    k= 1 3   As  k   is negative, P   divides  AB   in the ratio  1:3   externally.
    AP PB = 1 3   AP AB+AP = 1 3 3AP=AB+AP 2AP=AB AP= 1 2 AB  
    Therefore, the condition is AP= 1 2 AB  .
    Hence, option 1) is correct.
     
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