If Im i z + 2 z + i = - 1 represents part of a circle with radius r units, then the value of 4 r 2 is (where, z ∈ C ,   z ≠ i ,   Im z represents the imaginary part of z and i 2 = - 1 )

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 ⇒ the radius r of the circle is 1 4 2 + 1 2 = 3 4 Hence, 4 r 2 = 4 × 9 16 = 9 4 = 2 . 25