MathematicsPoint P   divides the line segment joining the points A(2,1)   and B(5,−8)   such that AP AB = 1 3 .   If P   lies on the line 2x−y+k=0  , then the value of k   from the following choices is:

Point P   divides the line segment joining the points A(2,1)   and B(5,8)   such that AP AB = 1 3 .   If P   lies on the line 2xy+k=0  , then the value of k   from the following choices is:


  1. A
    -2
  2. B
    -4
  3. C
    -6
  4. D
    -8 

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    Solution:

    Given that, the points are A(2, 1) and B(5, -8) and AP AB = 1 3 .  
    P   lies on the line 2xy+k=0  .
    If P(x, y) divides the line segment with the coordinates ( x 1 , y 1 ),( x 2 , y 2 )   in the ratio m:n internally, then the section formula is given by,
    P=( m x 2 +n x 1 m+n ),( m y 2 +n y 1 m+n )  
    We have to find the ratio for AP:PB.
    AP AB = 1 3 AP AP+PB = 1 3 3AP=AP+PB 2AP=BP   So, AP BP = 1 2  .
    Here, m = 1, n = 2.
    ( x 1 , y 1 )=(2,1) ( x 2 , y 2 )=(5,8)  
    By using section formula, the coordinate of x is given by,
    x= m x 2 +n x 1 m+n x= 1 5 +2 2 1+2 x=3   By using section formula, the coordinate of y is given by,
    y= m y 2 +n y 1 m+n y= 1 8 +2 1 1+2 y= 8+2 3 y=2  
    Substitute the value x=3,y=2   in the line equation, we get,
    2xy+k=0 2 3 (2)+k=0 6+2+k=0 k=8  
    The value of k is -8.
    Hence, option 4) is correct.
     
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