# Thread: Complex Numbers and Ellipse

1. ## Complex Numbers and Ellipse

Using the fact that |z1-z2| is the distance between two points z1 and z2, give a geometric argument that |z-4i|+|z+4i|=10 represents an ellipse whose foci are (0,4) and (0,-4). Can you help me get started, please?

2. ## Re: Complex Numbers and Ellipse

The [n]defining[/b] property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is a constant.

3. ## Re: Complex Numbers and Ellipse

Do I need to use the fact that z=x+iy?

4. ## Re: Complex Numbers and Ellipse

No! You could but the problem says "give a geometric argument". Just look closely at the problem! What points in the complex plane do "i " and "-i " correspond to?

5. ## Re: Complex Numbers and Ellipse

They correspond to (0,1) and (0,-1), so the distance between those two points is 2. The distance from each of those points to the foci is 3.

6. ## Re: Complex Numbers and Ellipse

Originally Posted by lovesmath
Using the fact that |z1-z2| is the distance between two points z1 and z2, give a geometric argument that |z-4i|+|z+4i|=10 represents an ellipse whose foci are (0,4) and (0,-4). Can you help me get started, please?
Because an ellipse is the set of all points such that the sum of there distances to two fixed points is constant.

How does that definition apply to $\displaystyle |z+4i|+|z-4i|=10~?$ How does it make the equation an ellipse ?
What are the two points in question?

7. ## Re: Complex Numbers and Ellipse

The two points in question are (0, 4i) and (0, -4i). The distance between those points is 8, so I don't understand where the 10 comes from. Am I interpreting the question incorrectly?

8. ## Re: Complex Numbers and Ellipse

Originally Posted by lovesmath
The two points in question are (0, 4i) and (0, -4i). The distance between those points is 8, so I don't understand where the 10 comes from. Am I interpreting the question incorrectly?
With all due respect, you real problem is that you have no idea what an ellipse really is.

You might study this webpage.

For your ellipse the focii are $\displaystyle (0,4)~\&~(0,-4)$,. Thus the major axis is vertical.

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# solve |z-4i| |z 4i|=10

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