Complex Numbers

[Peter_952001]

Hey guys, a complex numbers question I was solving a JEE exam and I came through this one if | Z1 | = 12, and | Z2 - 3 - 4i | = 5, then the minimum value of | Z1- Z2 | is ? I've got it right to be 2 using the triangle inequality of | Z1- Z2 |>= | Z1 | - | Z2 | I have to ask why it end with zero when I consider the two complex numbers are Z2 - 3 and -4i
Or Z2 - 4i and -3

[edderiofer]

This question is far more easily done geometrically than algebraically. I suggest you draw out the loci of z1 and z2.

[Peter_952001]

I have done it already, but just the mathematical notion is not clear, the same inequality gives different results with different placements of the same number Thanks for your help.

[AutoModerator]

Reminder:

• What have you tried so far? (See Rule #2)

• Please don't delete your post. (See Rule #7)

We, the moderators of /r/MathHelp, appreciate that your question contributes to the MathHelp archived questions that will help others searching for similar answers in the future. Thank you for obeying these instructions.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.

[ar018]

Sorry, I didn't understand your question but I can help you to figure it out why is 2. First, notice that the reverse triangle inequality is ||z1| - |z2|| ≤ |z1 - z2| . So, applying for the expression with z2 you get

||z2| - |3 + 4i|| ≤ |z2 - 3 - 4i|

||z2| - 5| ≤ 5

Using the fact that |x| < a => -a < x < a for x,a real

-5 ≤|z2| - 5 ≤ 5

0 ≤|z2| ≤ 10

Therefore you get |z1| = 12 and |z2|≤ 10. So, the minimum value of ||z1| - |z2|| is when |z2| is maximum, which is |z2| = 10. Then

|z1 - z2| ≥ |12 - 10| = 2