The point

P
divides the line segment joining the points

A(−1,3)
divides the line segment joining the pointsand

B(9,8)
andsuch that

AP

BP

=
k
1

such that. If

P
. If lies on the line

x−y+2=0
lies on the line ,then the value of k is ____.

Question

The point

P
  divides the line segment joining the points

A(−1,3)
  divides the line segment joining the pointsand

B(9,8)
 andsuch that

AP
    
     BP
   
   =
    k
    1

such that. If

P
 . If  lies on the line

x−y+2=0
  lies on the line ,then the value of k is ____.

Accepted Answer

The point

P
  divides the line segment joining the points

A(−1,3)
 and

B(9,8)
 such that

AP
    
     BP
   
   =
    k
    1

. If

P
  lies on the line

x−y+2=0
 ,then the value of k is 23.We have been given,The point

P
  divides the line segment joining the points

A(−1,3)
 and

B(9,8)
 such that

AP
    
     BP
   
   =
    k
    1

. Point

P
  lies on the line

x−y+2=0
 .Let A, B, and P be the three points and the coordinates of point A and B be

x
      1
     
     ,
      y
      1

and

x
      2
     
     ,
      y
      2

respectively. Using the section formula, (x, y)=mx2+nx1m+n, my2+ny1m+nLet point P divides A and B in a ratio k:1, Thus, the coordinates of P will be,

k
      x
      2
     
     +
      x
      1

k+1
   
   ,
    
     k
      y
      2
     
     +
      y
      1

k+1

We know,

A
    
     −1,3
   ,B
    
     9,8
   
 .Substituting m=k, x2=9, n=1, x1=1, y2=8, and y1=3 in the section formula,Now using section formula to find the coordinates of P,

k×9−1
    
     k+1
   
   ,
    
     k×8+3
    
     k+1
   
   =
    
     9k−1
    
     k+1
   
   ,
    
     8k+3
    
     k+1

So,

9k−1
    
     k+1
   
   =x and 
    
     8k+3
    
     k+1
   
   =y
  As we know, P lies on

x−y+2=0
 .Thus, substituting the value of x and y in the equation. ∴x-y+2=0Substituting the values,⇒9k-1k+1-8k+3k+1+2=0⇒9k-1-8k-3+2k+2k+1=0⇒3k-2=0

∴k=
    2
    3

Therefore, the value of ‘k’ is 23.

The point $P$ divides the line segment joining the points $A (− 1, 3)$ divides the line segment joining the pointsand $B (9, 8)$ andsuch that $A P B P = k 1$ such that. If $P$ . If lies on the line $x − y + 2 = 0$ lies on the line ,then the value of k is ____.

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The point P divides the line segment joining the points A(−1,3) divides the line segment joining the pointsand B(9,8) andsuch that AP BP = k 1 such that. If P . If lies on the line x−y+2=0 lies on the line ,then the value of k is ____.

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Detailed Solution

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Get Expert Academic Guidance – Connect with a Counselor Today!

best study material, now at your finger tips!

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The point $P$ divides the line segment joining the points $A (− 1, 3)$ divides the line segment joining the pointsand $B (9, 8)$ andsuch that $A P B P = k 1$ such that. If $P$ . If lies on the line $x − y + 2 = 0$ lies on the line ,then the value of k is ____.