‌AB=8 ‌AB2=64 ⇒(a−b)2+(b+4)2=64 Now P divides AB in the ratio 2 : 1 internally ⇒h=‌
2a+b
3
and k=‌
−4+b+2
3
⇒‌‌2a+b=3h k=‌
b−2
3
From equation (2) and (3) ‌‌⇒b=3k+2 ‌⇒‌‌2a=3h−3k−2 ‌⇒‌‌a=‌
3h−3k−2
2
Now by putting value of a and b in equation ‌⇒(‌
3h−3k−2
2
−(3k+2))2+(3k+2+4)2=64 ‌⇒(‌
3h−3k−2−6k−4
2
)2+(3k+6)2=64 ‌⇒(3h−9k−6)2+4(3k+6)2=4×64 ‌⇒9(h−3k−2)2+36(k+2)2=256 ‌⇒9(h2+9k2+4−6hk−4h+12k) ‌+36(k2+4+4k)=256 ‌⇒9(h2+13k2+20−6hk−4h+28k)=256 Replacing h by x and k by y ‌⇒9(x2+13y2−6xy−4x+28y)+180−256=0 ‌⇒9(x2+13y2−6xy−4x+28y)−76=0 By comparing α=13,β=−6,γ=−4 α−β−γ=13+6+4=23