Question

If (b + c), (c + a), (a + b) are in H.P., then which of the following hold(s) good?

• A. $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in A.P.
• B. $\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in H.P.
• C. $a^2,b^2,c^2$ are in A.P.
• D. $a^2,b^2,c^2$ are in H.P.

Answer 1

Answer: (B), (C)

The correct options are
B

$\frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in H.P.

C

$a^2,b^2,c^2$ are in A.P.

Given $\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}$ are in A.P. .....(i)

Multiplying by (a + b)(b + c)(c + a), we get

$\Rightarrow (a+b)(c+a),(a+b)(b+c),(b+c)(c+a)$ are in A.P.

$\Rightarrow a^2+(ab+bc+ca),b^2+(ab+bc+ca),c^2+(ab+bc+ca)$ are in A.P.

$\Rightarrow a^2,b^2,c^2$ are in A.P.

Again, by multiplying equation (i) by (a + b + c)

We get

$\Rightarrow \frac{a}{b+c}+1,\frac{b}{c+a}+1,\frac{c}{a+b}+1$ are in A.P.

$\Rightarrow \frac{b+c}{a},\frac{c+a}{b},\frac{a+b}{c}$ are in H.P.