Question

In $△$ ABC, if cot A, cot B, cot C be in A. P. then $a^2,b^2,c^2$ are in

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

Answer 1

The correct option is C

A.P.

cot A, cot B and cot C are in A.P.
$\Rightarrow$ cot A + cot C = 2 cot B $\Rightarrow \frac{cosA}{sinA}+\frac{cosC}{sinC}=\frac{2cosB}{sinB}$
$\Rightarrow \frac{b^2+c^2-a^2}{2bc\left(ka\right)}+\frac{a^2+b^2-c^2}{2ab\left(kc\right)}=2\frac{a^2+c^2-b^2}{2ac\left(kb\right)}$
$\Rightarrow a^2+c^2=2b^2.\text{Hence}{a}^2,b^2,c^2\text{are in A. P.}$

It might help to remember this result as a general relation.