JEE Advanced 2011 Paper 1Question 1
If $z$ is any complex number satisfying $|z-3-2i|\le2$, then the minimum value of $|2z-6+5i|$ is
Complex Numbers
Answer$5$
Solution
$|2z-6+5i|=2\left|z-\left(3-\tfrac52 i\right)\right|$. $z$ lies in the disc of centre $3+2i$, radius $2$; the distance from the centre to $3-\frac52i$ is $\frac92>2$, so the minimum of $|z-(3-\frac52 i)|$ is $\frac92-2=\frac52$. Minimum value $=2\cdot\frac52=5$.