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
Given that, R(x,y) divides PQ in the ratio k:1
Then we have,
R(X,Y)=(kx1+x2k+1,ky1+y2k+1)
Here, x1=a,y1=b, x2=b,y2=a
Then P(x,y)=(bk+ak+1,ak+bk+1)
⇒x=bk+ak+1 and y = (ak+bk+1)
⇒kx+x=bk+a and yk + y = ak + b
⇒k(x−b)=a−x ⇒k(y−a)=b−y
⇒k=a−xx−b ---(i) ⇒k = (b−yy−a) ---(ii)
from (i) and (ii)
a−xx−b=b−yy−a
⇒ay−a2−xy+ax=bx−b2+by−xy
⇒(a−b)y+(a−b)x−(a2−b2)=0
⇒(a−b)[y+x−(a+b)]=0
⇒x+y−(a+b)=0
⇒x+y=a+b
Hence proved