If $\mathrm{\alpha }$ and $\mathrm{\beta }$ are roots of the equation ${\mathrm{x}}^2-3\mathrm{x}+\mathrm{a}=0$ and $\gamma$ and $\mathrm{\delta }$ are roots of the  equation ${\mathrm{x}}^2-12\mathrm{x}+\mathrm{b}=0$ and $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma },\mathrm{\delta }$ form an increasing  GP, then the values  of a and b are respectively

$\because \mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma },\mathrm{\delta }$ are in GP.

Let $\mathrm{\alpha }=\mathrm{A},\mathrm{\beta }=\mathrm{Ar},\mathrm{\gamma }={\mathrm{Ar}}^2,\mathrm{\delta }={\mathrm{Ar}}^3$

$\because$$\mathrm{\alpha }$ and $\mathrm{\beta }$ are the roots of the equation ${\mathrm{x}}^2-3\mathrm{x}+\mathrm{a}=0$,
then  

$\begin{array}{r}\mathrm{\gamma }+\mathrm{\delta }=12\\ \Rightarrow {\mathrm{Ar}}^2(1+\mathrm{r})=12 ....\left(\mathrm{ii}\right)\end{array}$

On solving Eqs. (i) and (ii), we get

    $\mathrm{A}=1,\mathrm{r}=2$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{\alpha }=1,\mathrm{\beta }=2,\mathrm{\gamma }=4,\mathrm{\delta }=8\\ \therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{a}=\mathrm{\alpha \beta }=1\times 2=2\\ \text{ and }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{b}=\mathrm{\gamma \delta }=4\times 8=32\end{array}$