Equation of ellipse: complex plane

[jackscholar]

Homework Statement

The question simply states that the focals are (0,2) and (2,-1) and I need to form an equation from it. I know that in complex form this would be |z-(0-2i)| + |z-(-2+i)| or more simply |z+2i|+|z+2-i|. Is this right?

 

[Simon Bridge]

You should always show your reasoning.
How many points do you need to know to uniquely specify a conic section?

http://www.ams.org/bookstore/pspdf/amstext-12-prev.pdf

 

[jackscholar]

Well, in my textbook the examples always say "describe the locus of z(x,y) given that... and there will be an example such as the |z-2i|+|z+1|=4 the problem with what I've done is that it doesn't have an axis length and I don't know if I need one or how to find one.

 

[Simon Bridge]

Have you covered conic sections in the Cartesian plane?
How many points do you need to know to uniquely specify a conic section?

It looks to me like you are applying formulas without understanding them.

Note:
f=(2,-1) means f=2-i right?
So: |z-(-2+i)| = |z+(2-i)| = |z+f| ... is what you wrote isn't it?

 

[HallsofIvy]

Homework Statement

The question simply states that the focals are (0,2) and (2,-1) and I need to form an equation from it. I know that in complex form this would be |z-(0-2i)| + |z-(-2+i)| or more simply |z+2i|+|z+2-i|. Is this right?

"I know in complex form this would be". What would be? Grammatically, the only thing "this" could apply to is "an equation" but you don't have an equation!

Yes, if z is a point on the ellipse, the |z+ 2i| and |z+ 2- i| are the distances from that point to the two foci. The geometric definition of an ellipse is "The sum of the distances from any point on the ellipse to the foci is a constant." So you should have, not just "|z+ 2|+ |z+ 2+ i|" but that equal to some number.

 

[jackscholar]

That is what I wrote in regards to Simon Bridge. To uniquely specify a conic section you need five points, don't you? How do I determine the distances between those two points? The constant is equal to the major axis length, isn't it. In determining the major axis length I would be able to determine that |z+2|+|z+2+i|= whatever the number is?

 

[Simon Bridge]

That is what I wrote in regards to Simon Bridge.

It's what I was hoping to jog your memory towards, yes.

To uniquely specify a conic section you need five points, don't you?

Any three points on the curve will do.
I think you need to give yourself a refresher on conic sections.

Does the information in the problem statement you have written uniquely specify the ellipse?

How do I determine the distances between those two points? The constant is equal to the major axis length, isn't it.

You can easily check that by sketching an ellipse, and the line segments between each focus and a point where the ellipse crosses the major axis.

In determining the major axis length I would be able to determine that |z+2|+|z+2+i|= whatever the number is?

Are you given any clues to the major axis length, or to the location of any point on the ellipse?