NEED HELP ASAP!....please :)

[Moko]

I came across this qs in chapter 4 ER essential maths (qs16.b)

it says:     Let S={z: l z - (2(sqr root2) + i 2(sqr root2) l (less than or equal to -->)2

         a) If z belongs to S, find the max and min values of lzl
       

         b) If z belongs to S, find the max and min values of Arg (z)

Pls explain fully

THanks

[oneoneoneone]

What that expression represents is a circle with radius 2 and center at 2rt2 + i 2rt2.
So the maximum value of |z| would be the point on that circle that is furthest from the origin, and the minimum value would be the point on that circle thats closest to the origin. I believe this corresponds to |3rt2 + i 3rt2| and |rt2 + i rt2|, which are 6 and 2 respectively.

For part b, the Argument of z is the angle between the line drawn from z to the origin and the positive direction of the x axis taken anticlockwise. Hence the maximum value of Arg z would be Arg ( rt2 + i 3rt2) and the minimum value of Arg z would be Arg(3rt2 + i rt2)

[Moko]

For the first part you're right, but can you just explain how you got to "|3rt2 + i 3rt2| and |rt2 + i rt2|, which are 6 and 2 respectively"?

[oneoneoneone]

What i did was sketch the reigon defined by the Question, then the point with the smallest value of |z| will be the point that is directly between 2rt2 + i 2rt2 and the origin. Since we know its 2 units away (as the radius of the circle is 2), we know the modulus will be 2 less than the modulus of the center of the circle. |2rt2 + i 2rt2| = rt(4*2 + 4*2)=rt(16)=4 hence the  smallest modulus will be 1. Likewise, the point with the largest value of |z| will be the point on the line between the origin and 2rt2 + i 2rt2 that is not between the origin and 2rt2 + i 2rt2, so its modulus will be 2 greater than |2rt2 + i 2rt2| and it is hence 6.

That seems really confusing reading over it so ill upload a picture if you need it.

[Moko]

Lol yeah it actually is,
If you have time to post a pic that would be great 

[oneoneoneone]

Link

I hope thats a bit clearer :\ The main thing is

Draw the Region
See where the maximum and minimum value of |z| is the maximum and minimum distance from the origin in the region.
Notice that both these points lie on the line between the centre of the circle and the origin
Since the radius is 2, each point on the edge of the circle will be 2 units away from the centre
This distance from the closest point on the circle to the origin will be 2 less than that of the center of the circle to the origin, so the minimum modulus would be 2.

[Moko]

Thanks a lot man that really helps