Question

The largest value of $r$ for which the region represented by the set $\{\omega \in C:|\omega -4-i|\le r\}$ is contained in the region represented by the set $\{z\in C:|z-1|\le |z+i|\},$ is equal to:

• A. $\frac{3}{2}\sqrt{2}$ unit
• B. $\sqrt{17}$ unit
• C. $2\sqrt{2}$ unit
• D. $\frac{5}{2}\sqrt{2}$ unit

Answer 1

The correct option is D $\frac{5}{2}\sqrt{2}$ unit

$\{\omega \in C:|\omega -4-i|=r\}$ represents a circle with centre at $(4,1)$ and radius $r$.
$|\omega -4-i|\le r\Rightarrow$ All points lying inside circle.
$\{z\in C:|z-1|=|z+i|\}$ represents the line $y=-x$
$|z-1|\le |z+i|\Rightarrow$ region consisting of point $z=1$
$(\because |1-1|\le |1+i|)$
$\therefore$ Required region is internal to circle
$r$ will be largest if the given line is tangent to the circle.


So, $r$ will be the perpendicular distance between centre $(4,1)$ and the line $y=-x$.
$r_{max}=\frac{4+1}{\sqrt{1^2+1^2}}=\frac{5}{\sqrt{2}}=\frac{5}{2}\sqrt{2}$ unit