Question

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

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

Answer 1

Answer: (A)

(i) and (ii) both correct

Answer 2

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