If R( x , y ) is a point on the line segment joining the points P( a , b ) and Q( b , a ) , then prove that x+y =a+b

Since R(x, y) is a point on the line segment joining the point, P(a, b) and Q(b, a).

∴ P(a, b), Q(b, a) and R(x, y)are the collinear.

⇒ Area of ΔPQR = 0

Area of triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) .

 

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P : ( a,b )

Q : ( b,a )

R : ( x,y )

let the point R devide line segment PQ in k :1

so x = bk + a / k + 1 ....(i)

also y = ak + b / k + 1 ...(ii)

x + y = bk + a/ (k+1) + ak + b/(k+1)

  = bk + a + ak + b/ k+1

  = a + b + (a + b)k / k+1

  = a + b { 1 + k / k + 1} = a + b

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