Question

If $\alpha ,\beta$ are the roots of equation $x^2-4x+\lambda =0$ and $\gamma ,\delta$ are the roots of the equation $x^2-64x+\mu =0$ and $\alpha ,\beta ,\gamma ,\delta$ are given to be in increasing $G.P$, find the values of $\lambda$ and $\mu$.

Answer 1

$\alpha +\beta =4,\alpha \beta =\lambda ,y+\delta =64,\gamma \delta =\mu$
$\alpha ,\beta ,\gamma ,\delta$ are in increasing G.P $\therefore r>1$
$\beta =\alpha r,\gamma =\alpha r^2,\delta =\alpha r^3$
$\alpha (1+r)=4,{\alpha }^{2}r=\lambda$
$\alpha r^2(1+r)=64,{\alpha }^2{r}^5=\mu$
Dividing $r^2=16$ $\therefore r=4$ only as $r>1$
$\therefore \alpha =\frac{4}{5}$ $\therefore \lambda =a^{2}r=\frac{16}{25}.4=\frac{64}{25}$
$\mu ={\alpha }^2.r^5={\left(\frac{4}{5}\right)}^2.4^5=\frac{4^2.4^5}{5\times 5}=\frac{4^7}{25}$
Hence $\lambda =\frac{64}{25}$, $\mu =\frac{4^7}{25}$