Is it straight forward?

By properties of conjugate: $z_1\overline {z_1}=5^2, z_2\overline {z_2}=13^2$ . Thus, $\begin{aligned} 39z_1-15z_2&=\frac {39}{13^2}z_1z_2\overline {z_2}-\frac {15}{5^2}z_2z_1\overline {z_1}\\&=z_1z_2(\frac {3}{13}\overline {z_2}-\frac {3}{5}\overline {z_1})\\&=z_1z_2(\frac {15\overline {z_2}-39\overline {z_1}}{65})\\&=-\frac {z_1z_2}{65}\overline {39z_1-15z_2}\end{aligned}$ .

Rearranging the equation gives us: $\begin{aligned}z_1z_2&=-65\frac {39z_1-15z_2}{\overline {39z_1-15z_2}}\\&=-65\frac {4+7i}{4-7i}\\&=-(4+7i)^2\\&=33-56i\end{aligned}$ . Therefore, $a=33,b=-56$ and $2a+b=\boxed {10}$