# Thread: Subset of the complex plane

1. ## Subset of the complex plane

I need help on another:

|z-4-i|+|z+4+i| > 1

2. Looks like the "stuff" outside an ellipse.

3. Originally Posted by damo17
I need help on another:

|z-4-i|+|z+4+i| > 1
First find |z - 4 - i| + |z + 4 + i| = 1 to get the boundary.

Note that this can be written as |z - (4 + i)| + |z - (-4 - i)| = 1.

Now ..... if you're familiar with the locus definition of an ellipse then there's a very simple way of seeing that there's no solution to this equation.

Therefore the answer to the inequality is the entire complex plane ....

4. Originally Posted by manjohn12
Looks like the "stuff" outside an ellipse.
The equality does not define an ellipse because the distance between the focal points z = 4 + i and z = -4 - i is greater than 1 ....

5. Originally Posted by mr fantastic
The equality does not define an ellipse because the distance between the focal points z = 4 + i and z = -4 - i is greater than 1 ....

So it is $\sqrt{68} > 1$. Oh okay. Was thinking the equality was equal to 10.

6. Originally Posted by damo17
I need help on another:

|z-4-i|+|z+4+i| > 1
In your (unnecessary) re-posting of this question you said:

Originally Posted by damo17
I need help on another question:

|z-4-i|+|z+4+i| = 1

[snip]
So what's it meant to be then, an inequality or an equality? Either way, my first post contains the answer.

If you need an algebraic approach where you substitute z = x + iy etc. please say so.

7. Thanks,
sorry for any confusion it's >1.
I'm still confused a bit. Could you show working out by substituting z=x+yi in. Thanks