Question

If $r_{1},r_{2},r_{3}$in a triangle be in H.P., then the sides are in

• A. A.P
• B. G.P
• C. H.P
• D. None of these

Answer 1

Answer: (A)

A.P

Answer 2

$\frac{1}{r_{2}} - \frac{1}{r_{1}} = \frac{1}{r_{3}} - \frac{1}{r_{2}}$

⇒ $\frac{1}{\frac{\Delta}{s - b}} - \frac{1}{\frac{\Delta}{s - a}} = \frac{1}{\frac{\Delta}{s - c}} - \frac{1}{\frac{\Delta}{s - b}}$ $$\Rightarrow \frac{s - b - s + a}{\Delta} = \frac{s - c - s + b}{\Delta} \Rightarrow \frac{a - b}{\Delta} = \frac{b - c}{\Delta}$$

⇒ $\frac{b - a}{\Delta} = \frac{c - b}{\Delta}$ ⇒ $2b = a + c$.