Question - Find the minimum value of $|1 + z| + |1-z|$.
I'm trying to solve the question by thinking of them as points in the Argand plane. The $|1+z|$ can be written as $|z - (-1)|$ which is the distance of $z$ from $(-1) $ on the Argand plane. But I don't understand how to find the second part on Argand plane like I did the first one. If I find the second point, then the answer will just be the minimum distance between both the points.
On
$|1+z|+|1-z|=c$ for a positive constant $c>0$ dscribes an ellipse with focus at $1$ and $-1$ and axis of length $c$.
BEcause the axis must be longer than the distance between focus, $c\ge 2$.
On
A bit of geometry.
In the complex plane:
$A(-1,0)$, $B(1,0)$ and let $C(x,y),$ where $z =x+iy$, $x,y$, real.
$\triangle ABC$ has sides of lengths:
$|AB|=2$, |$AC| =|z+1|$, $|BC|= |z-1|$.
The sum of the lengths of 2 sides of a triangle is greater than the 3rd side:
$|z+1|+|z-1| > 2.$
$2$ is a lower bound. Is there a minimum?
If yes , $z=?$
The formula represents the sum of the distances from z to 1 and -1. So the minimum is at the midpoint of -1 , 1, i. e. 0; thus the minimum value is 2.