Weird question (1 Viewer)

[oompaman]

Was doing some past papers and this came up

Find the least value of |z| when |z-4-3i| = 3
I tried doing it with triangular inequalities and got a negative soln (got -2 instead of 2)

I can do it via an argand diagram, but i can't seem to find a fault in my algebra work.

help?

 

[jyu]

Was doing some past papers and this came up

Find the least value of |z| when |z-4-3i| = 3
I tried doing it with triangular inequalities and got a negative soln (got -2 instead of 2)

I can do it via an argand diagram, but i can't seem to find a fault in my algebra work.

help?

|z-4-3i| = 3 is a circle of radius 3, and its centre is 5 units from (0,0). So the closest z is 2 units from (0,0).
Show your algebra work?

 

[oompaman]

|z-4-3i| = 3 is a circle of radius 3, and its centre is 5 units from (0,0). So the closest z is 2 units from (0,0).
Show your algebra work?

Kk, so
Using triangular inequalities |z-(4+3i)| <= |z| + |4+3i|
3 <= |z| + 5
so |z| >= 3-5
|z| >= -2

Which gives a min value of -2 rather than 2?
Unless i made a mistake somewhere.

 

[Rezen]

Kk, so
Using triangular inequalities |z-(4+3i)| <= |z| + |4+3i|
3 <= |z| + 5
so |z| >= 3-5
|z| >= -2

Which gives a min value of -2 rather than 2?
Unless i made a mistake somewhere.

Your algebra is right, you're just misinterpreting what the equations are telling you. What you have proven is that for any z on the circle, its modulus >=-2. It's not saying that for some z on the circle, its modulus is -2 (a nonsensical statement since modulus is always positive).

Edit: If you use the reverse triangle inequality, you can infact prove what the maximal and minimal |z| is.

 

[oompaman]

Your algebra is right, you're just misinterpreting what the equations are telling you. What you have proven is that for any z on the circle, its modulus >=-2. It's not saying that for some z on the circle, its modulus is -2 (a nonsensical statement since modulus is always positive).

Edit: If you use the reverse triangle inequality, you can infact prove what the maximal and minimal |z| is.

so what i have done is find something useless?
Also what do you mean by reverse triangle inequality?

 

[Rezen]

so what i have done is find something useless?
Also what do you mean by reverse triangle inequality?

It doesn't give you any extra information so I guess you could say that. The reverse triangle inequality is the identity . From this when can derive bounds on |z|.


and


Then it's only a matter of proving that |z| takes these two values.

 

[P.T.F.E]

It is just a circle with radius of 3, center (4,3) on Argand diagram