If R (x,y) is a point on the line segment joining the point P ( a ,b)

1.	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» e 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

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»

e 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