Clearly for the shaded region z1 is the intersection of the circle and the line passing through P(L1) and z2 is intersection of line L1 & L2 Circle : (x+2)2+(y−3)2=1 ‌L1:x+y−1=0 ‌L2:x−y+4=0 On solving circle &L1 we get z1:(−2−‌
1
√2
,3+‌
1
√2
) On solving L1 and z2 is intersection of line L1&L2 we get z2:(‌
−3
2
,‌
5
2
) |z1|2+2|z2|2=14+5√2+17 ‌=31+5√2 ‌ So ‌‌α=31 ‌β=5 ‌α+β=36