Question

If a complex number $z$ lie on a circle of radius $\frac{1}{2}$ units, then the complex number $\omega =-1+4\text{z}$ will always lie on a circle of radius $k$ units, where $k$ is equal to

Answer 1

Answer: 2

Let us assume that $z$ lies on a circle with centre $z_0$ (fixed point) and radius $\frac{1}{2}$ units.
$\Rightarrow \left|\text{z}-{\text{z}}_0\right|=\frac{1}{2}$
Now, $\omega =-1+4\text{z}\Rightarrow \omega +1=4\text{z}$
$\Rightarrow \omega +1-4{\text{z}}_0=4\text{z}-4{\text{z}}_0$
Now, taking modulus on both sides, we get,
$\left|\omega +1-4{\text{z}}_0\right|=4\left|\text{z}-{\text{z}}_0\right|\Rightarrow \left|\omega +1-4{\text{z}}_0\right|=2$
Locus of $\omega$ represents the circle having centre $\left(-1+4z_0\right)$ and radius $2$ units.