If-b-a-b-3a-b-c-2b-c-a-3c-a-2a-b-3c-a-b-c-0-then-show-that-a-b-c-

Question Number 202540 by MATHEMATICSAM last updated on 29/Dec/23

If \frac{b}{a + b} = \frac{3 a - b - c}{2 b + c - a} = \frac{3 c - a}{2 a - b + 3 c}

[a + b + c \neq 0] then show that a = b = c .

Answered by som(math1967) last updated on 29/Dec/23

\frac{b}{a + b} = \frac{3 a - b - c}{2 b + c - a} = \frac{3 c - a}{2 a - b + 3 c}

\frac{3 b}{3 a + 3 b} = \frac{3 a - b - c}{2 b + c - a} = \frac{3 c - a}{2 a - b + 3 c}

∴ e a c h r a t i o

= \frac{3 b + 3 a - b - c + 3 c - a}{3 a + 3 b + 2 b + c - a + 2 a - b + 3 c}

= \frac{2 (a + b + c)}{4 (a + b + c)} = \frac{1}{2} [(a + b + c) \neq 0]

∴ \frac{b}{a + b} = \frac{1}{2} \Rightarrow 2 b = a + b ∴ a = b

a g a i n \frac{3 c - a}{2 a - b + 3 c} = \frac{1}{2}

\Rightarrow 6 c - 2 a = 2 a - a + 3 c [∵ a = b]

\Rightarrow 3 c = 3 a ∴ c = a

∴ a = b = c