Question

if the point R(x,y) is a point lying on the line segment joining the points P(a,b) and Q(b,a) prove that x+y= a+b.​

Answer 1

Answer:

Given that, R(x,y) divides PQ in the ratio k:1

Then we have,

R(X,Y)=(

k+1

kx

1

+x

2

,

k+1

ky

1

+y

2

)

Here, x

1

=a,y

1

=b, x

2

=b,y

2

=a

Then P(x,y)=(

k+1

bk+a

,

k+1

ak+b

)

⇒x=

k+1

bk+a

and y = (

k+1

ak+b

)

⇒kx+x=bk+a and yk + y = ak + b

⇒k(x−b)=a−x ⇒k(y−a)=b−y

⇒k=

x−b

a−x

---(i) ⇒k = (

y−a

b−y

) ---(ii)

from (i) and (ii)

x−b

a−x

=

y−a

b−y

⇒ay−a

2

−xy+ax=bx−b

2

+by−xy

⇒(a−b)y+(a−b)x−(a

2

−b

2

)=0

⇒(a−b)[y+x−(a+b)]=0

⇒x+y−(a+b)=0

⇒x+y=a+b

Hence proved

Step-by-step explanation:

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Answer 2

Step-by-step explanation:

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