Question

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

Answer 1

Slope of the both the segments must be same(as it lies on the same line).

Slope{(using (a, b) and (b, a)] is (b - a)/(a - b).

Slope[using (x, y) and (b, a)] is (b - y)/(a - x).

    Both must be equal:

 ∴ (b - a)/(a - b) = (b - y)/(a - x)

⇒ (b - a)(a - x) = (b - y)(a - b)

⇒ ab - bx - a² + ax = ab - b² - ay + by

⇒ - bx - a² + ax = - b² - ay + by

⇒ b² - a² - bx + ax = - ay + by

⇒ (b + a)(b - a) - x(b - a) = y(b - a)

⇒ (b - a)[ (b + a) - x) ] = (b - a)y

⇒ ( b + a - x ) = y

⇒ a + b = x + y

  Proved

Answer 2

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SEE ATTACHMENT

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hope it helps