Question

If m & M denotes the minimum and maximum value of $|2z+1|$ where$|z-2i|\le 1$ then $(m+M{)}^2$ is equal to

• A. $17$
• B. $34$
• C. $51$
• D. $68$

Answer 1

The correct option is D $68$

$|2z+1|=?$

Where $|z-2i|\le 1,$ then ${(m+M)}^2$

$\Rightarrow |2z+1|=2|z+\frac{1}{2}|.$ We can denote this as $2x,$

Where $x$ is at a distance of $z$ from $\frac{1}{2}.$

The distance is minimum if $z$ lies on the line joining the point.

$z$ lies on the left or right of $z.$ Let both $x$ and $y$ be positive then $y=\frac{1}{2}+x.$

$\Rightarrow z=\frac{1}{2}x=\frac{1}{2}\times 16=8$

$\Rightarrow |2\times 8+1|=\sqrt{17}=m$

$\therefore |z-2i|\le 1$

$\sqrt{z^2-4i-4}\le 1$

$z^2-4i-4\le 1$

from this $z=4=M.$

$\Rightarrow {(M+m)}^2={(4+\sqrt{17})}^2=68.$

Hence, the answer is $68.$