Question

If $\alpha ,\beta$ are the roots of the equation $x^2-4x+\lambda =0$ and $\gamma ,\delta$ are the roots of the equation $x^2-64x+\mu =0$ and $\alpha ,\beta ,\gamma ,\delta$ forms an increasing G.P., then the value of $\frac{\mu }{\lambda }$ is

Answer 1

Assuming $\alpha =a,\beta =ar,\gamma =a{r}^2,\delta =a{r}^3$
$x^2-4x+\lambda =0$
So,
$\alpha +\beta =4\alpha \cdot \beta =\lambda$
$x^2-64x+\mu =0$
So,
$\gamma +\delta =64\gamma \cdot \delta =\mu$
Now,
$\frac{\gamma +\delta }{\alpha +\beta }=16\Rightarrow r^2=16\Rightarrow r=\pm 4$
Therefore,
$\frac{\mu }{\lambda }=\frac{\gamma \cdot \delta }{\alpha \cdot \beta }\Rightarrow \frac{\mu }{\lambda }=r^4\therefore \frac{\mu }{\lambda }=256$