If a complex number z lie on a circle of radius 1 2 units, then the complex number ω = - 1 + 4 z will always lie on a circle of radius k units, where k is equal to

Step-by-step solution

Let us assume that z lies on a circle with centre z 0 (fixed point) and radius 1 2 units. ⇒    z - z 0 = 1 2 Now, ω = - 1 + 4 z ⇒ ω + 1 = 4 z ⇒    ω + 1 - 4 z 0 = 4 z - 4 z 0 Now, taking modulus on both sides, we get, ω + 1 - 4 z 0 = 4 z - z 0 ⇒ ω + 1 - 4 z 0 = 2 Locus of ω represents the circle having centre - 1 + 4 z 0 and radius 2 units.