Let z be a complex such that:
1<=|z|<=6.
Then what are the minimum and maximum values for the complex (z+3).
I suspect that:
0<=|z+3|<=9, but I can't find a way to prove it.
Please help!
Moo, I already tried that method and came to the same result.
I thought of it in a different way - the proof isn't at all rigorous; in fact it's just an insight not a proof.
By adding 3 to z we are shifting the point representing z (on the complex plane) 3 units to the right.
Now since z can be -3, this gives us |z+3|=0. So this has to be the minimum - the smallest possible value for a modulus being 0.
Also if z = 6, then |z+3| = 9, and you will find that whatever other point z you take, the modulus |z+3| will always be smaller than 9.
But as I said, this is only an insight. I can't get any further.
Do you see something that I don't?
This is what the problem wants to say:
Find such |z| that and that maximizes (or minimizes) |z+3|.
Now this is only an optimization question =)
will make minimum. So,
For maximum,
The maximum x and y values that satisfy and maximize are 6 and 0.
This is how to find it:
Maximize:
Maximize:
There is no term of y in 6x + 9 so x must be maximum. Then and