Question

$\alpha ,\beta$ be the roots of the equation $x^2-3x+a=0$ and $\gamma ,\delta$ the roots of $x^2-12x+b=0$ and numbers $\alpha ,\beta ,\gamma ,\delta$ (in this order) form an increasing G.P, then

• A. $a=3,b=12$
• B. $a=12,b=3$
• C. $a=2,b=32$
• D. $a=4,b=16$

Answer 1

The correct option is C $a=2,b=32$

Let $\alpha =x$

$\beta =x\times r$
$\gamma =x\times r^2$

$\delta =x\times r^3$

As sum of roots is $\frac{-b}{a}$ and product of roots is $\frac{c}{a}$ in $a{x}^2+bx+c$

So $\alpha +\beta =3$

$x+xr=3$

$x(1+r)=3$ (1)

Similarly $x{r}^2+x{r}^3=12$

$x{r}^2(1+r)=12$ (2)

Dividing (2) by (1)

$r^2=4$

$r=2$ or $r=-2$

Taking $r=2$ and substituting in (1)

$x(1+2)=3$

$x=1$

So,

$\alpha =1$

$\beta =2$

$\gamma =4$

$\delta =8$

So $a=1\times 2$ and $b=4\times 8$

Hence $a=2$ and $b=32$

So option (C) is correct answer.