# Thread: Sketching |z - 1| + |z + i| <= 2 in complex plane

1. ## Sketching |z - 1| + |z + i| <= 2 in complex plane

Sketch the following region in the complex plane: $\displaystyle { z \epsilon\mathbb{C} : |z -1| + |z + i| \leq 2}$

Attempt:

$\displaystyle |z - 1| + |z + i| \leq 2$

$\displaystyle \sqrt{(x - 1)^2 + y^2} + \sqrt{x^2 + (y + 1)^2} \leq 2$

$\displaystyle \sqrt{(x - 1)^2 + y^2} \leq 2 - \sqrt{x^2 + (y + 1)^2}$

$\displaystyle x^2 - 2x + 1 + y^2 \leq 4 - 4\sqrt{x^2 + y^2 + 2y + 1} + (x^2 + y^2 + 2y + 1)$

$\displaystyle -2x - 4 - 2y \leq -4\sqrt{x^2 + y^2 + 2y + 1}$

$\displaystyle (-2x - 4 - 2y)^2 \leq 16(x^2 + y^2 + 2y + 1)$

$\displaystyle 4x^2 + 16x + 16y + 8xy + 16 + 4y^2 \leq 16(x^2 + y^2 + 2y + 1)$

It looks like I have reached a dead end

2. $\displaystyle |z-1|$ is the distance of z to 1.
$\displaystyle |z+i|$ is the distance of z to -i.
So you have the sum of two is less than or equal 2.
Think ellipse.

3. Or: -12x^2+8xy+16 x-12y^2-16y <0

Which is an ellipse!

4. Originally Posted by SyNtHeSiS
Sketch the following region in the complex plane: $\displaystyle { z \epsilon\mathbb{C} : |z -1| + |z + i| \leq 2}$

Attempt:

$\displaystyle |z - 1| + |z + i| \leq 2$

$\displaystyle \sqrt{(x - 1)^2 + y^2} + \sqrt{x^2 + (y + 1)^2} \leq 2$

$\displaystyle \sqrt{(x - 1)^2 + y^2} \leq 2 - \sqrt{x^2 + (y + 1)^2}$

$\displaystyle x^2 - 2x + 1 + y^2 \leq 4 - 4\sqrt{x^2 + y^2 + 2y + 1} + (x^2 + y^2 + 2y + 1)$

$\displaystyle -2x - 4 - 2y \leq -4\sqrt{x^2 + y^2 + 2y + 1}$

$\displaystyle (-2x - 4 - 2y)^2 \leq 16(x^2 + y^2 + 2y + 1)$

$\displaystyle 4x^2 + 16x + 16y + 8xy + 16 + 4y^2 \leq 16(x^2 + y^2 + 2y + 1)$

It looks like I have reached a dead end
The boundary of that set is given by $\displaystyle 4x^2 + 16x + 16y + 8xy + 16 + 4y^2 = 16(x^2 + y^2 + 2y + 1)$
which, as Plato and also Sprach Zarathustra have told you, is an ellipse. The set you want is the interior and boundary of that set (because of the "<").

5. Thanks. How would you change

$\displaystyle -12x^2 + 16x - 12y^2 - 16y + 8xy \leq 0$

into the standard form on an ellipse:

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ?

6. The quick answer is: one does not.
The axes are not vertical/horizontal.
The foci are (1,0) and (0,-1)

7. In that case, how would you know that it is an ellipse, with only knowing those 2 points (1,0) and (0, -1)?

8. Originally Posted by SyNtHeSiS
In that case, how would you know that it is an ellipse, with only knowing those 2 points (1,0) and (0, -1)?
ELLIPSE
The path of a point which moves so that the sum of its distances from two fixed points is a constant. The two fixed points are the foci of the ellipse.

9. The internal region is where the sum is <2.

Sorry about the typo on the sketch, should be |z+i|.

10. 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?

11. Originally Posted by SyNtHeSiS
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?
The notation $\displaystyle \left| {z - z_0 } \right|$ is the distance from $\displaystyle z$ to $\displaystyle z_0$.

Therefore $\displaystyle \left| {z - z_0 } \right|+\left| {z - z_1 } \right|=C$ must be an ellipse.

12. Originally Posted by Plato
The notation $\displaystyle \left| {z - z_0 } \right|$ is the distance from $\displaystyle z$ to $\displaystyle z_0$.

Therefore $\displaystyle \left| {z - z_0 } \right|+\left| {z - z_1 } \right|=C$ must be an ellipse.
Provided $\displaystyle C > \left|z_1 - z_0\right|$.

@OP: If $\displaystyle C = \left|z_1 - z_0\right|$ then the locus is a line segment joining $\displaystyle z_0$ and $\displaystyle z_1$, and if $\displaystyle C < \left|z_1 - z_0\right|$ then there is no locus.