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

Answer 1

Answer: (C)

$\frac{5}{2}\sqrt{2}$

The region represented by the set {$\omega \in C/|\omega -4-i|\le r$} is a circular region whose center is $(4,1)$
$(x-4{)}^2+(y-1{)}^2\le r^2$ ------(1)
And the region represented by the set {$z\in C/|z-1|\le |z+i|$} is
$(x-1{)}^2+y^2\le x^2+(y+1{)}^2$
$\Rightarrow -2x+1\le 2y+1$
$\Rightarrow x+y\ge 0$ -----(2)
Limiting condition is $x+y=0$ is tangent to the circle $(x-4{)}^2+(y-1{)}^2=r^2$
$\Rightarrow \frac{4+1}{\sqrt{2}}=r$
$\Rightarrow r=\frac{5}{\sqrt{2}}=\frac{5\sqrt{2}}{2}$
$\therefore$ Largest value of $r$ is $\frac{5\sqrt{2}}{2}$
Hence, option C.