Weird question (1 Viewer)

oompaman

Member
Joined
Apr 29, 2012
Messages
85
Gender
Undisclosed
HSC
N/A
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

Member
Joined
Nov 14, 2005
Messages
623
Gender
Male
HSC
2006
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?
 
Last edited:

oompaman

Member
Joined
Apr 29, 2012
Messages
85
Gender
Undisclosed
HSC
N/A
|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

Member
Joined
Mar 12, 2009
Messages
62
Gender
Male
HSC
2010
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.
 
Last edited:

oompaman

Member
Joined
Apr 29, 2012
Messages
85
Gender
Undisclosed
HSC
N/A
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

Member
Joined
Mar 12, 2009
Messages
62
Gender
Male
HSC
2010
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

Member
Joined
Nov 6, 2008
Messages
333
Gender
Male
HSC
2013
It is just a circle with radius of 3, center (4,3) on Argand diagram
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top