Question

If a(y+z)=b(z+x)=c(x+y) and out of a, b, c no two of them are equal then show that, y-z/a(b-c)=z-x/b(c-a)=x-y/c(a-b)

Answer 1

given, a(y + z) = b(z + x) = c(x + y) where out of a, b, c ; no two of them are equal.

now, a(y + z) = b(z + x) = c(x + y) = k (let)

so, a(y + z) = k ⇒(y + z) = k/a ......(1)

similarly, b(z + x) = k ⇒(z + x) = k/b.....(2)

c(x + y) = k ⇒(x + y)= k/c .....(3)

now equation (2) - equation (1),

(z + x) - (y + z) = k/b - k/a

⇒(x - y) = k (a - b)/ab

⇒(x - y)/(a - b) = k/ab

⇒(x - y)/c(a - b) = k/abc ......(4)

similarly, equation (3) - equation (2),

(y - z)/a(b - c) = k/abc ......(5)

equation (1) - equation (3)

(z - x)/b(c - a) = k/abc ......(6)

from equations (4), (5) and (6),

(x - y)/c(a - b) = (y - z)/a(b - c) = (z - x)/b(c - a)

⇒(y - z)/a(b - c) = (z - x)/b(c - a) = (x - y)/c(a - b)