Answer:
Option C
Explanation:
Given, $\alpha$ and $\beta$ are the roots of the equation $x^{2}-6x-2=0$
$\therefore$ $a_{n}=\alpha^{n}-\beta^{n}$ $n\geq 1$
$\therefore$ $a_{10}=\alpha^{10}-\beta^{10}$
$a_{8}=\alpha^{8}-\beta^{8}$
$a_{9}=\alpha^{9}-\beta^{9}$
Now, consider
$\frac{a_{10}-2a_{8}}{2a_{9}}=\frac{\alpha^{10}-\beta^{10}-2(\alpha^{8}-\beta^{8})}{2(\alpha^{9}-\beta^{9})}$
$=\frac{\alpha^{8}(\alpha^{2}-2)-\beta^{8}(\beta^{2}-2))}{2(\alpha^{9}-\beta^{9})}$
$=\frac{\alpha^{8}.6\alpha-\beta^{8}6\beta}{2(\alpha^{9}-\beta^{9})}$
$=\frac{6\alpha^{9}.-6\beta^{9}}{2(\alpha^{9}-\beta^{9})}=\frac{6}{2}=3$
{ $ \therefore $ $\alpha $ and $\beta$ are the roots of the equation:
$x^{2}-6x-2=0$
or $\alpha^{2}$ =6 $\alpha$ +2
$\Rightarrow$ $\alpha^{2}-2=6\alpha$
and $\beta^{2}=6\beta-2$
$\Rightarrow$ $\beta^{2}-2=6\beta$ }
Alter :
Since , $\alpha$ and $\beta$ be thr roots of equation $x^{2}-6x-2=0$
or $x^{2}=6x+2$
$\therefore$ $\alpha^{2}$ =6 $\alpha$ +2
$\Rightarrow$ $\alpha^{10}=6\alpha^{9}+2\alpha^{8}$ ......(i)
Similarly, $\beta^{10}=6\beta^{9}+2\beta^{8}$.................(ii)
On subtracting Eq. (ii) from Eq.(i) , we get
$\alpha^{10}-\beta^{10}=6(\alpha^{9}-\beta^{9})+2(\alpha^{8}-\beta^{8})$ ( $\Rightarrow a_{10}=6a_{9}+2a_{8} (\because a_{n}=\alpha^{n}-\beta^{n})$ )
$\Rightarrow a_{10}-2a_{8}=6a_{9}\Rightarrow \frac{a_{10}-2a_{8}}{2a_{9}}=3$
