= Solution
Under equal costs for either wrong assignment, the <Bayes classifier> chooses the class of greatest <posterior probability>. With positive prior probabilities, it assigns class one when $\pi_1f_1(x)>\pi_2f_2(x)$; ties can be resolved arbitrarily. For the two <multivariate normal densities> with common positive-definite <covariance matrix> $V$, their normalizing factors cancel. The log posterior odds are
$$
\begin{aligned}
\log\frac{\pi_1f_1(x)}{\pi_2f_2(x)}
&=\log\frac{\pi_1}{\pi_2}-\frac12\big[(x-\mu_1)^TV^{-1}(x-\mu_1)-(x-\mu_2)^TV^{-1}(x-\mu_2)\big]\\
&=(\mu_1-\mu_2)^TV^{-1}x-\frac12(\mu_1^TV^{-1}\mu_1-\mu_2^TV^{-1}\mu_2)+\log\frac{\pi_1}{\pi_2}.
\end{aligned}
$$
Therefore the <Gaussian Bayes classifier> has the required linear form, with
$$
\boxed{a=V^{-1}(\mu_1-\mu_2),\qquad b=\frac12(\mu_1+\mu_2)^TV^{-1}(\mu_1-\mu_2)-\log\frac{\pi_1}{\pi_2}.}
$$
A larger prior probability for class one decreases $b$, expanding its decision region. The <prior-dependent Gaussian discriminant boundary> is the hyperplane $a^Tx=b$.
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