# Complex Numbers and Ellipse

• July 3rd 2013, 07:35 PM
lovesmath
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?
• July 3rd 2013, 07:54 PM
HallsofIvy
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.
• July 4th 2013, 10:09 AM
lovesmath
Re: Complex Numbers and Ellipse
Do I need to use the fact that z=x+iy?
• July 4th 2013, 12:38 PM
HallsofIvy
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?
• July 4th 2013, 02:01 PM
lovesmath
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.
• July 4th 2013, 02:15 PM
Plato
Re: Complex Numbers and Ellipse
Quote:

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 $|z+4i|+|z-4i|=10~?$ How does it make the equation an ellipse ?
What are the two points in question?
• July 7th 2013, 02:03 PM
lovesmath
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?
• July 7th 2013, 02:41 PM
Plato
Re: Complex Numbers and Ellipse
Quote:

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 $(0,4)~\&~(0,-4)$,. Thus the major axis is vertical.