Finding value in a complex set region

[Raghav Gupta]

Homework Statement

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

Homework Equations

complex number = a + ib where a,b ε R

The Attempt at a Solution

Don't know how to start or what to apply

 

[mooncrater]

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

 

[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?

 

[mooncrater]

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

 

[Raghav Gupta]

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

 

[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...

 

[mooncrater]

Is the D option 5√2/2 or 5/2√2?

 

[Raghav Gupta]

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

What's the line equation?
Is it y = -x ?
And circle equation is (x-4)² + (y-1)² = r² ?

Is the D option 5√2/2 or 5/2√2?

The D option is 5√2/2 .

 

[mooncrater]

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

Yes.

 

[Raghav Gupta]

Got it, thanks the D option.