Finding value in a complex set region

1. Apr 21, 2015

Raghav Gupta

1. The problem statement, all variables and given/known data

The largest value of r for which the region represented by the set { ω ε C / |ω - 4 - i| ≤ r}
is contained in the region represented by the set { z ε C / |z - 1| ≤ |z + i|}, is equal to :
√17
2√2
3/2 √2
5/2 √2
2. Relevant equations
complex number = a + ib where a,b ε R

3. The attempt at a solution
Don't know how to start or what to apply

2. Apr 21, 2015

mooncrater

Hi!!!
Hint: the set |w-4-i| ≤r represents the region inside the circle with its centre (4, 1) and radius r.

3. Apr 21, 2015

Raghav Gupta

Hello Mooncrater
Okay, we know centre of circle as (4,1) and radius r.
Now taking z= x + iy,
we get from question
(x - 1)2 + y2 ≤ x2+ (y+1)2
⇒ -x ≤ y
Now what?

4. Apr 21, 2015

mooncrater

Now each equation(the circle and the line) points out where a point can be.. like a constraint.

5. Apr 21, 2015

Raghav Gupta

So should we differentiate to get max. Value of r?
Is it a minima and maxima problem?

6. Apr 21, 2015

mooncrater

You can do it through graph... it will be very easier then. I think maxima would work if you want it to do it that way...

7. Apr 21, 2015

mooncrater

Is the D option 5√2/2 or 5/2√2?

8. Apr 21, 2015

Raghav Gupta

What's the line equation?
Is it y = -x ?
And circle equation is (x-4)2 + (y-1)2 = r2 ?
The D option is 5√2/2 .

9. Apr 21, 2015

mooncrater

Yes.

10. Apr 22, 2015

Raghav Gupta

Got it, thanks the D option.