⇒bxdy+cydy+ady=ax‌dx−by‌dx+a‌dx ⇒bxdy+by‌dx+(cy+a)dy=(ax+a)‌dx ⇒b(xdy+y‌dx)+(cy+a)dy=(ax+a)‌dx ⇒bd(xy)+(cy+a)dy=(ax+a)‌dx Integrating both sides, we get ⇒b‌∫d(xy)+∫(cy+a)dy=∫(ax+a)‌dx ⇒b⋅xy+c⋅‌
y2
2
+ay=‌
ax2
2
+ax+k ⇒‌
ax2
2
−‌
cy2
2
−bxy+ax−ay+k=0 For equation of circle, Coefficient of x2= Coefficient of y2 ∴‌
a
2
=−‌
c
2
⇒a=−c And coefficient of xy=0 ∴−b=0 ⇒b=0 ∴ Circle equation becomes, ‌
ax2
2
+‌
ay2
2
+ax−ay+k=0 ⇒x2+y2+2x−2y+‌
2k
a
=0 ∴‌ Center ‌=(−g,−f)=(−1,1)=(α,β) ∴α=−1‌ and ‌β=1 ∴α+2β=−1+2×(1)=1