Question

lf $|z|=3$, then the points representing the complex numbers $-1+4z$ lie on a

• A. $line$
• B. $circle$
• C. $parabola$
• D. $hyperbola$

Answer 1

The correct option is B $circle$

$|z|=3\Rightarrow z=3(\cos \theta +i\sin \theta )\left(1\right)$
$x+iy=-1+4z\left(2\right)$
Now from (1) and (2),
$x+iy=-1+12(\cos \theta +i\sin \theta )$
Comparing both side,
$x=-1+12\cos \theta$
and $y=12\sin \theta$
$\Rightarrow x+1=12\cos \theta$ and $y=12\sin \theta$
Now take square of $(x+1)$ and $y$, then add
$\Rightarrow (x+1{)}^2+y^2=144$ is a circle