Given, $r_1=2, r_2=3$ and $r_3=6$
$$
\begin{aligned}
& \because \quad r_1=\frac{\Delta}{s-a} \Rightarrow 2=\frac{\Delta}{s-a} \\
& \Rightarrow \frac{s-a}{\Delta}=\frac{1}{2} \\
& r_2=\frac{\Delta}{s-b} \Rightarrow 3=\frac{\Delta}{s-b} \\
& \Rightarrow \quad \frac{s-b}{\Delta}=\frac{1}{3} \\
& \text { and } \\
& r_3=\frac{\Delta}{s-c} \Rightarrow 6=\frac{\Delta}{s-C} \\
& \Rightarrow \quad \frac{s-c}{4}=\frac{1}{6} \\
&
\end{aligned}
$$
On adding Eqs. (i), (ii) and (iii), we get
$$
\begin{aligned}
& \frac{s-a}{\Delta}+\frac{(s-b)}{4}+\frac{s-c}{\Delta}=\frac{1}{2}+\frac{1}{3}+\frac{1}{6} \\
\Rightarrow & \frac{3 s-(a+b+c)}{4}=\frac{3+2+1}{6} \\
\Rightarrow & \frac{3 s-2 s}{4}=\frac{6}{6}=0 \\
\Rightarrow & \frac{s}{\Delta}=1 \\
\because & s^2=r_1 r_2+r_2 r_3+r_3 r_1
\end{aligned}
$$
$$
\begin{aligned}
& =2 \times 3+3 \times 6+6 \times 2 \\
& =6+18+12=36 \\
\Rightarrow \quad s^2 & =36 \Rightarrow s=6
\end{aligned}
$$
From Eq. (i), we get
$$
\begin{aligned}
\frac{6}{\Delta} & =1 \\
\text { Now, } \quad \quad \quad \quad \quad r_1 & =\frac{\Delta}{s-a} \\
2 & =\frac{6}{6-a} \\
\Rightarrow \quad 6-a & =3 \Rightarrow a=3
\end{aligned}
$$