Question

If $z$ is any complex number satisfying $|z-3-2i|\le 2$, then the minimum value of $|2z-6+5i|$ is

Answer 1

$|z-3-2i|\le 2$
$\Rightarrow z$ lies on or inside the circle radius $2$ and centre $(3,2)$


Now,
$|2x-6+5i{|}_{min}=2{|z-3+\left(\frac{5}{2}\right)i|}_{min}$

$=2\times \left({}_{\text{on the given circle to the point}}^{\text{minimum distance of any point}}(3,-\frac{5}{2})\right)$

$=2\times BA$

$=2\left(\frac{5}{2}\right)=5$