Answer :

**Given :** R (x, y) is a point on the line segment joining the points P (a, b) and Q (b, a).**To prove:** x + y = a + b**Proof:**

It is said that the point

*R*(

*x, y*) lies on the line segment joining the points

*P*(

*a, b*) and

*Q*(

*b, a*). Thus, these three points are collinear.

So the area enclosed by them should be 0.

Area of the triangle having vertices (x_{1},y_{1}), (x_{2},y_{2}) and (x_{3},y_{3}) is:

Given that area of ∆PQR = 0

∴ |x(b – a) + a(a – y) + b(y – b)| = 0

∴ bx – ax + a^{2} - ay + by - b^{2} = 0

∴ ax + ay –bx – by - a^{2} - b^{2} = 0

∴ ax + ay –bx – by = a^{2} + b^{2}

(a – b)(x + y) = (a – b )(a + b)

∴ x +y = a + b

Hence proved.

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