Question

The sum of the squares of three distinct real numbers, which are in G .P., is $S^2$. If their sum is $\alpha S$ then answers the following questions.${\alpha }^2$ lies-

• A. $(1/3,2)$
• B. $(1,2)$
• C. $(1/2,3)$
• D. None of these

Answer 1

The correct option is D None of these

Let terms of G.P. are $\frac{a}{r},a,ar$

Given $\frac{a}{r}+a+ar=\alpha s$

or $a\left(\frac{1+r+r^2}{r}\right)=\alpha s.....\left(i\right)$

and $\frac{a^2}{r^2}+a^2+a^2{r}^2=s^2$

or $a^2\left(\frac{r^4+r^2+1}{r^2}\right)=s^2....\left(ii\right)$

On squaring (i) & dividing by (ii)

We get .$\frac{r^2+r+1}{r^2-r+1}={\alpha }^2.....\left(iii\right)$

by (iii), $r^2(1-{\alpha }^2)+r(1+{\alpha }^2)+(1-{\alpha }^2)=0$

Since increasing $G.P.\Rightarrow D>0$

$\frac{1}{3}<{\alpha }^2<3$.