Im⁡2z+1iz+1=5 represents

12i2z+1iz+1−2z¯+1−iz¯+1=5⇒ (2z+1)(iz¯−1)+(2z¯+1)(iz+1) =10i(iz+1)(iz¯−1)⇒ 4izz¯+(−2+i)z+(2+i)z¯ =10i(−zz¯−iz+iz¯−1)⇒ 14izz¯+(8+i)z+(−8+i)z¯−10i=0⇒ zz¯+114(1−8i)z+114(1+8i)z¯−57=0This represents a circle