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

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 $\left(-1+4 z _{0}\right)$ and radius $2$ units.