If Im i z - 2 z - i = - 1 , (where z ∈ C , z ≠ i ) represents part of a circle, then the radius (in units) of the circle is (here, Im ( z ) represents the imaginary part of the complex number z )

Step-by-step solution

Let, z = x + i y I m i x + y i - 2 x + i y - i = - 1 ⇒ I m - y + 2 + i x x + i y - 1 = - 1 ⇒ I m - y + 2 + i x x - i y - 1 x 2 + y - 1 2 = - 1 ⇒ x 2 + y 2 + y - 2 = - x 2 - y 2 + 2 y - 1 ⇒ 2 x 2 + 2 y 2 - y - 1 = 0 ⇒ x 2 + y 2 - y 2 - 1 2 = 0 ⇒ radius of the circle is 1 4 2 + 1 2 = 3 4 = 0 . 75