Sketch the following region in the complex plane:

Attempt:

It looks like I have reached a dead end(Speechless)

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- Oct 27th 2010, 03:18 AMSyNtHeSiSSketching |z - 1| + |z + i| <= 2 in complex plane
Sketch the following region in the complex plane:

Attempt:

It looks like I have reached a dead end(Speechless) - Oct 27th 2010, 03:42 AMPlato
is the distance of z to 1.

is the distance of z to -i.

So you have the sum of two is less than or equal 2.

Think ellipse. - Oct 27th 2010, 03:48 AMAlso sprach Zarathustra
Or: -12x^2+8xy+16 x-12y^2-16y <0

Which is an ellipse! - Oct 27th 2010, 08:44 AMHallsofIvy
- Oct 27th 2010, 10:07 AMSyNtHeSiS
Thanks. How would you change

into the standard form on an ellipse:

? - Oct 27th 2010, 10:14 AMPlato
The quick answer is: one does not.

The axes are not vertical/horizontal.

The foci are (1,0) and (0,-1) - Oct 27th 2010, 10:39 AMSyNtHeSiS
In that case, how would you know that it is an ellipse, with only knowing those 2 points (1,0) and (0, -1)?

- Oct 27th 2010, 10:47 AMPlato
- Oct 27th 2010, 10:55 AMArchie Meade
The internal region is where the sum is <2.

Sorry about the typo on the sketch, should be |z+i|. - Oct 28th 2010, 03:04 AMSyNtHeSiS
Thanks. I am still a bit confused. I understand that the definition of an ellipse is that the sum of the distances from both foci to a point is constant, but how would you know this in this example, without testing 2 points on the ellipse, to see if the sum of their distances from both foci are equal?

- Oct 28th 2010, 03:22 AMPlato
- Oct 28th 2010, 05:08 AMmr fantastic