Problem $2$ Power factor compensation $($UMF $8\mathrm{.}50$ modified$)$

a$)$ Find the power factor of the entire load seen by the voltage source including the resistor $R_1$ assuming there is no compensation capacitor.

b$)$ Find the value of the shunt capacitor required to make the power factor exactly $1.$

c$)$ Find the voltage $V_{ab}$ and the current supplied by the voltage source both with and without the compensation capacitor if the magnitude of the of source voltage is $100$ V $.$

$8\mathrm{.}50$ In the phasor$-$domain circuit shown in Fig. P$8\mathrm{.}50,R_1=1\mathrm{.}2k,R_2=5k,L_1=0\mathrm{.}8H,L_1=0\mathrm{.}6H,C=2F,$ and $=1000ra\frac{d}{s}.$