Question

if (a-b)/c +(b-c)/a+(c-a)/b=1 and (a-b+c) is not equal to zero then prove that 1/a=1/b+1/c

Answer 1

$\frac{1}{a} = \frac{1}{b} + \frac{1}{c}$ (Proved)

Step-by-step explanation:

Given that, $\frac{a - b}{c} + \frac{b - c}{a} + \frac{c - a}{b} = 1$

⇒ ab(a - b) + bc(b - c) + ca(c - a) = abc

⇒ a²b - ab² + b²c - bc² + c²a - ca² = abc

⇒ (a²b + a²c - abc) - (b²a + abc - b²c) + (abc + ac² - bc²) = 0

⇒ a(ab + ac - bc) - b(ab + ac - bc) + c(ab + ac - bc) = 0

⇒ (a - b + c)(ab + ac - bc) = 0

⇒ ab + ac - bc = 0 {Since, (a - b + c) ≠ 0}

⇒ ab + ac = bc

⇒ $\frac{1}{a} = \frac{1}{b} + \frac{1}{c}$ (Proved)