Question

If $b^2,a^2,c^2$ are in $AP$ then $a+b,b+c,c+a$ will be in

• A. $AP$
• B. $GP$
• C. $HP$
• D. $Noneofthese$

Answer 1

The correct option is C $HP$

$a^2,b^2,c^2$ are in AP

So, $b^2-a^2=c^2-b^2$

$\Rightarrow 2b^2=c^2+a^2$ …………..$\left(1\right)$

as $a^2,b^2,c^2$ are in AP

So, $\frac{a}{a^2},\frac{1}{b^2},\frac{1}{c^2}$ are in HP

$\Rightarrow b^4=a^2{c}^2$

$\Rightarrow b^2=ac$ ……………$\left(2\right)$

From $\left(1\right)$ & $\left(2\right)$

$2\times ac=c^2+a^2$

$\Rightarrow a^2+c^2-2ac=0$

$\Rightarrow (a-c{)}^2=0$

$\therefore a=c$ …………$\left(3\right)$

From $\left(3\right)$ & $\left(2\right)$

$b^2=c^2$

$\Rightarrow b=c$ ………..$\left(4\right)$

So, $a=b=c$

$\Rightarrow (b+c{)}^2=(a+b\left)\right(c+a)$.

They are in HP.