Question

If $a^2,b^2,c^2$ are in A.P. consider two statements
(i) $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ are in A.P.
(ii) $\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in A.P.

• A. (i) and (ii) both correct
• B. (i) and (ii) both incorrect
• C. (i) and (ii) both incorrect
• D. (i) incorrect (ii) correct

Answer 1

The correct option is A (i) and (ii) both correct

Given $a^2,b^2.c^2$ are in A.P.
$\Rightarrow a^2+(ab+bc+ca),b^2+(ab+bc+ca),c^2+(ab+bc+ca)\text{are in A.P.}$
$\Rightarrow (a+b)(a+c),(b+c)(b+a),(c+a)(c+b)$ are in A.P.
$\Rightarrow \frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ are in A.P.
$\left[Divideby\right(a+b\left)\right(b+c\left)\right(c+a\left)\right]$
Again, $a^2,b^2,c^2$ are in A.P.
$\Rightarrow \frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ are in A.P.
$\Rightarrow \frac{a+b+c}{b+c},\frac{a+b+c}{c+a},\frac{a+b+c}{a+b}$ are in A.P.
$\Rightarrow \frac{a}{b+c}+1,\frac{b}{c+a}+1,\frac{c}{a+b}+1$ are in A.P.
$\Rightarrow \frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}$ are in A.P.