Question

If $\left|z\right|=2$, then the points representing the complex numbers $-1+5z$ will lie on a

• A. circle
• B. straight line
• C. parabola
• D. None of these

Answer 1

The correct option is A

circle

Explanation for the correct option.

Form an equation.

Let $z_1=x+iy$ be a complex number that lies on $-1+5z$, then

$$\begin{gathered} z_1=-1+5z \\ \Rightarrow z+iy+1=5z \\ \Rightarrow \left(x+1\right)+iy=5z \end{gathered}$$

Now, take the modulus of both sides and form an equation.

$$\begin{gathered} \left|\left(x+1\right)+iy\right|=\left|5z\right| \\ \Rightarrow \sqrt{{\left(x+1\right)}^2+y^2}=5\left|z\right| \\ \Rightarrow \sqrt{{(x+1)}^2+y^2}=5\times 2\left[\left|z\right|=2\right] \\ \Rightarrow {\left(x+1\right)}^2+y^2={10}^2 \end{gathered}$$

This equation ${\left(x+1\right)}^2+y^2={10}^2$ represents a circle whose center is $\left(-1,0\right)$ and radius is $10$.

So, the points representing the complex numbers $-1+5z$ will lie on a circle.

Hence, the correct option is A.