Complex numbers2

**thyristor** · March 21st, 2010, 12:02 PM

Hi!
This is a problem taken from a textbook, but it's not a homework problem. I'm just rehearsing for a test.
Determine which complex numbers satisfy the equation

.
You are supposed to make a geometrical interpretation, and in the key it says "the complex numbers that lie on the ellipse

.
However, I don't understand how they arrive at this result. Could somebody explain, please?

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**Leszek Luchowski** · March 21st, 2010, 12:41 PM

is the Euclidean distance, in the complex plane, between

and

Correction: not -1 but -i. Likewise,

is the Euclidean distance between

and

. Correction: not 1 but i.

Now the locus of points

such that the sum of the distances between

and two given points is equal to a given constant (greater than the distance between the two points) is an ellipse.

You do the math to transform the equation to the form shown in your answer. Hint: use the Pythagorean theorem to determine the distances without using the absolute value symbol.

Good luck, and keep us posted.

**thyristor** · March 21st, 2010, 01:53 PM

Originally Posted by Leszek Luchowski

is the Euclidean distance, in the complex plane, between

and

. Likewise,

is the Euclidean distance between

and

.

Now the locus of points

such that the sum of the distances between

and two given points is equal to a given constant (greater than the distance between the two points) is an ellipse.

You do the math to transform the equation to the form shown in your answer. Hint: use the Pythagorean theorem to determine the distances without using the absolute value symbol.

Good luck, and keep us posted.

I'm sorry if I don't understand, but why would

be the distance between z and -1?

**mathman** · March 21st, 2010, 03:42 PM

It looks like these are typos. The distances are to -i and i respectively. The original equation means that the sum of the distances to these two points = 3. The geometrical definition of an ellipse is of this form.