A $100$ km long three phase line has a sending$-$end voltage of $138$ kV and transmits half the maximum real power. The line reactance is $80\frac{}{?}$ phase.

$($a$)$ Find $$ and $|V_2|.$

$($b$)$ The line is operated with $138$ kV in both ends with $100$ MW flow in direction $12.$ Find power angle and reactive power flow in line ends.