Question

If$a,bandc$ be in H.P, then

• A. $\frac{a-b}{b-c}=\frac{a}{c}$
• B. $\frac{b-c}{c-a}=\frac{b}{a}$
• C. $\frac{c-a}{a-b}=\frac{c}{b}$
• D. $\frac{a-b}{b-c}=\frac{c}{a}$

Answer 1

The correct option is A

$\frac{a-b}{b-c}=\frac{a}{c}$

Finding the value:

Given: $a,bandc$ be in H.P

$\therefore \frac{1}{a},\frac{1}{b}and\frac{1}{c}$are in A.P

, $$\begin{gathered} \frac{1}{b}-\frac{1}{a}=\frac{1}{c}-\frac{1}{b} \\ \Rightarrow \frac{a-b}{ba}=\frac{b-c}{cb} \\ \Rightarrow \frac{a-b}{b-c}=\frac{ba}{cb} \\ \Rightarrow \frac{a-b}{b-c}=\frac{a}{c} \end{gathered}$$

Hence option$\mathbf{\left(}\mathit{A}\mathbf{\right)}$ is correct.