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Question: If *a*, *b*, *c* are in H.P., then which one of the following is true...

If a, b, c are in H.P., then which one of the following is true

A

1b−a+1b−c=1b\frac{1}{b - a} + \frac{1}{b - c} = \frac{1}{b}

B

aca+c=b\frac{ac}{a + c} = b

C

b+ab−a+b+cb−c=1\frac{b + a}{b - a} + \frac{b + c}{b - c} = 1

D

None of these

Answer

None of these

Explanation

Solution

a, b, c are in H.P. ⇒ b=2aca+cb = \frac{2ac}{a + c}, ∴ option (2) is false

b−a=2aca+c−a=a(c−a)c+ab - a = \frac{2ac}{a + c} - a = \frac{a(c - a)}{c + a} ⇒ b−c=c(a−c)a+cb - c = \frac{c(a - c)}{a + c}

∴ 1b−a+1b−c=a+ca−c{−1a+1c}=a+ca−c⋅a−cac=a+cac=a+c2ac⋅2=2b\frac{1}{b - a} + \frac{1}{b - c} = \frac{a + c}{a - c}\left\{ - \frac{1}{a} + \frac{1}{c} \right\} = \frac{a + c}{a - c} \cdot \frac{a - c}{ac} = \frac{a + c}{ac} = \frac{a + c}{2ac} \cdot 2 = \frac{2}{b}, ∴ option (1) is false

}{\left\{ - \left( \frac{b + a}{a} \right) + \frac{b + c}{c} \right\} = \frac{a + c}{a - c}\left( \frac{b}{c} - \frac{b}{a} \right) = \frac{a + c}{a - c} \cdot \frac{(a - c)b}{ac} = \frac{a + c}{ac} \cdot b = \frac{a + c}{2ac} \cdot 2b = \frac{1}{b} \cdot 2b = 2}$$ ∴ Option (3) is false.