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.

Question

Philoid · Accepted Answer

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 (x₁,y₁), (x₂,y₂) and (x₃,y₃)

= $\frac{1}{2}|x1(y2-y3)+x2(y3-y1)+x3(y1-y2)|$

Given that area of ∆PQR = 0

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

∴ bx – ax + a² - ay + by - b² = 0

∴ ax + ay –bx – by - a² - b² = 0

∴ ax + ay –bx – by = a² + b²

(a – b)(x + y) = (a – b )(a + b)

∴ x +y = a – b

Hence proved.