$1..$ The transfer function $G(s)$ in the block diagram is of the form: $G(s)=\frac{s+a}{s+b}$ and it has the following parameter values: $a=1,b=2.$ The values of the compensator transfer function $G_c(s)=\frac{s+z_c}{s+p_c}$ are unknown. However, the zero of the compensator is given as $z_c=-4\mathrm{.}15$

It is required that the closed loop loop system $T(s)$ has a pole at the point $s_0=-4\mathrm{.}15+j3\mathrm{.}15$ Use root locus techniques to determine the value of the pole $p_c$ of the compensator, such that the closed loop loop system has a pole at $s_0=-4\mathrm{.}15+j3\mathrm{.}15$

An equivalent way to phrase this requirement is to : design the compensator $G_c(s),$ so that the root locus for the system will pass through the point $s_0.$

Additionally, determine the value of the gain $K$ such that this requirement is satisfied.