Given that circle x2+y2+8x−4y+c=0 touch the circle x2+y2+2x+4y−11=0 then, C1C2=r1+r2 .....(i) where C1=(−4,2) r1=√16+4−c=√20−c C2=(−1,−2) and r2=√1+4+11=4 ∴ From Eq. (i), √(−4+1)2+(2+2)2 =√20−c+4 ⇒ 5=√20−c+4 ⇒ c=19 Also, the circle x2+y2+8x−4y+c=0 cuts the circles x2+y2−6x+8y+k=0 orthogonally, then c1→(−4,2) C3→(3,−4) (C1C3)2=(r1)2+(r3)2‌‌‌ where, r1=√16+4−c r3=√9+16−k ⇒ {(−4−3)2+(2+4)2}=(20−c)+(25−k) ⇒ 49+36=45−k−c ⇒ k+c=−40 ⇒ k+19=−40 ⇒ k=−59